J .  fi.  Ijto  tLcL 


IN  MEMORIAM 
FLORIAN  CAJORI 


THR 


THEORY  OF  SUBSTITUTIONS 


AND    ITS 


APPLICATIONS  TO  ALGEBRA. 


By  dr.  EUGEN  NETTO, 

Professor  of  Mathematics  in  the  Univeksity  of  Giessen. 


Revised   by  the  Author   and   Translated   with    his    Permission 
By  F.  N.  cole.  Ph.  D., 

Assistant  Professor  of  Mathematics  in    the 
f  University  of  Michigan. 


ANN  ARBOR,  MICH,: 

the  register  publishing  company. 

Ube  f  nlanb  prees. 


\t 


PREFACE. 

The  presentation  of  the  Theory  of  Substitutions  here  given  differs  in 
several  essential  features  from  that  which  has  heretofore  been  custom- 
ary. It  will  accordingly  be  proper  in  this  place  to  state  in  brief  the 
guiding  principles  adopted  in  the  present  work. 

It  is  unquestionable  that  the  sphere  of  application  of  an  Algorithm 
is  extended  by  eliminating  from  its  fundamental  principles  and  its 
general  structure  all  matters  and  suppositions  not  absolutely  essential 
to  its  nature,  and  that  through  the  general  character  of  the  objects  with 
which  it  deals,  the  possibility  of  its  employment  in  the  most  varied 
directions  is  secured.  That  the  theory  of  the  construction  of  groups 
admits  of  such  a  treatment  is  a  guarantee  for  its  far-reaching  impor- 
tance and  for  its  future. 

If,  on  the  other  hand,  it  is  a  question  of  the  application  of  an  aux- 
iliary method  to  a  definitely  prescribed  and  limited  problem,  the  elab- 
oration of  the  method  will  also  have  to  take  into  account  only  this 
one  purpose.  The  exclusion  of  all  superfluous  elements  and  the 
increased  usefulness  of  the  method  is  a  sufficient  compensation  for  the 
lacking,  but  not  defective,  generality.  A  greater  efficiency  is  attained  in 
a  smaller  sphere  of  action. 

The  following  treatment  is  calculated  solely  to  introduce  in  an 
elementary  manner  an  important  auxiliary  method  for  algebraic  inves- 
tigations. By  the  employment  of  integral  functions  from  the  outset,  it 
is  not  only  possible  to  give  to  the  Theory  of  Substitutions,  this  operat- 
ing with  operations,  a  concrete  and  readily  comprehended  foundation, 
but  also  in  many  cases  to  simplify  the  demonstrations,  to  give  the 
various  conceptions  which  arise  a  precise  form,  to  define  sharply  the 
principal  question,  and— what  does  not  appear  to  be  least  important — to 
limit  the  extent  of  the  work. 

The  two  comprehensive  treatises  on  the  Theory  of  Substitutions 
which  have  thus  far  appeared  are  those  of  J.  A.  Serret  and  of  C.  Jordan. 

The  fourth  section  of  the  " Alg^bre  Supdrieure  "  of  Serret  is  devoted 
to  this  subject.  T^^e  radical  difference  of  the  methods  involved  here 
and  there  hardly  permitted  an  employment  of  this  highly  deserving 
work  for  our  purposes.  Otherwise  Avith  the  more  extensive  work  of 
Jordan,  the  "Traits  des  substitutions  et  des  Equations  alg^briques." 
Not  only  the  new  fundamental  ideas  were  taken  from  this  book,  but  it 
js  proper  to  mention  expressly  here  that  many  of  its  proofs  and  pro- 


IV  PKEFACE. 

cesses  of  thought  also  permitted  of  being  satisfactorily  employed  in  the 
present  work  in  spite  of  the  essential  difference  of  the  general  treat- 
ment. The  investigations  of  Jordan  not  contained  in  the  "  Traits " 
which  have  been  consulted  are  cited  in  the  appropriate  places. 

But  while  many  single  particulars  are  traceable  to  this  "Traits" 
and  to  these  investigations,  nevertheless,  the  author  is  indebted  to  his 
honored  teacher,  L.  Kronecker,  for  the  ideas  which  lie  at  the  foundation 
of  his  entire  work.  He  has  striven  to  employ  to  best  advantage  the 
benefit  which  he  has  derived  from  the  lectures  and  from  the  study  of 
the  works  of  this  scholarly  man,  and  from  the  inspiring  personal  inter- 
course with  him;  and  he  hopes  that  traces  of  this  influence  may  appear 
in  many  places  in  his  work.  One  thing  he  regrets:  that  the  recent  im- 
portant publication  of  Kronecker,  "Grundztige  einer  arithmetischen 
Theorie  der  algebraischen  Grossen,"  appeared  too  late  for  him  to  derive 
from  it  the  benefit  which  he  would  have  wished  for  himself  and  his 
readers. 

The  plan  of  the  present  book  is  as  follows: 

In  the  first  part  the  leading  principles  of  the  theory  of  substitutions 
are  deduced  with  constant  regard  to  the  theory  of  the  integral  func- 
tions; the  analytical  treatment  retires  almost  wholly  to  the  background, 
being  employed  only  at  a  late  stage  in  reference  to  the  groups  of  solvable 
equations. 

In  the  second  part,  after  the  establishment  of  a  few  fundamental 
principles,  the  equations  of  the  second,  third  and  fourth  degrees,  the 
Abelian  and  the  Galois  equations  are  discussed  as  examples.  After  this 
follows  a  chapter  devoted  to  an  arithmetical  discussion  the  necessity  of 
which  is  there  explained.  Finally  the  more  general,  but  still  elementary 
questions  with  regard  to  solvable  equations  are  examined. 

Stkassburg,  1880. 


To  the  preceding  I  have  now  to  add  that  the  present  translation 
differs  from  the  German  edition  in  many  important  particulars.  Many 
new  investigations  have  been  added.  Others,  formerly  included,  which 
have  shown  themselves  to  be  of  inferior  importance,  have  been  omitted. 
Entire  chapters  have  been  rearranged  and  demonstrations  simplified. 
In  short,  the  whole  material  which  has  accumulated  in  the  course  of 
time  since  the  first  appearance  of  the  book  is  now  turned  to  account. 

In  conclusion  the  author  desires  to  express  his  warmest  thanks  to 
Mr.  Y.  N.  Cole  who  has  disinterestedly  assumed  th#task  of  translation 
and  performed  it  with  care  and  skijl, 

EUGEN  KETTO. 

CllESSEN,  1892, 


TRANSLATOE'S  NOTE. 


The  translator  has  confined  himself  almost  exclusively  to  the 
function  of  rendering  the  German  into  respectable  English.  My  thanks 
are  especially  due  to  The  Register  Publishing  Company  for  their  gener- 
ous assumption  of  the  expense  of  publication  and  to  Mr.  C.  N.  Jones,  of 
Milwaukee,  for  valuable  assistance  while  the  book  was  passing  through 

the  press. 

F.  N.  COLE. 

Ann  Arbor,  February  27,  1892. 


TABLE  OF   CONTENTS. 


PART  I. 

Theory  of  Substitutions  and  of  Integral  Functions. 

CHAPTER  I. 

SYMMETRIC    OR    SINGLE-VALUED   FUNCTIONS ALTERNATING    AND   TWO-VAL- 
UED   FUNCTIONS. 

1-3.  Symmetric  and  single- valued  functions. 

4 .  Elementary  symmetric  functions. 

5-10.  Treatment  of  the  symmetric  functions. 

11.  Discriminants. 

12 .  Euler's  formula. 

13.  Two- valued  functions;  substitutions. 

14 .  Decomposition  of  substitutions  into  transpositions, 

15.  Alternating  functions. 

16-20.  Treatment  and  group  of  the  two- valued  functions. 

CHAPTER  II. 

MULTIPLE-VALUED    FUNCTIONS    AND    GROUPS    OF    SUBSTITUTIONS. 

22.  Notation  for  substitutions. 

24.  Their  number. 

25.  Their  applications  to  functions. 
26-27 .  Products  of  substitutions. 

28 .  Groups  of  substitutions. 

29-32 .  Correlation  of  function  and  group. 

34.  Symmetric  group. 

35.  Alternating  group. 

36-38 .  Construction  of  simple  groups. 

39-40.  Group  of  order  p/. 

CHAPTER  III. 

THE    DIFFERENT    VALUES     OF    A    MULTIPLE-VALUED    FUNCTION    AND    THEIR 
ALGEBRAIC    RELATION    TO    ONE    ANOTHER. 

41-44 .  Relation  of  the  order  of  a  group  to  the  number  of  values 

of  the  corresponding  function. 


Till  CONTENTS. 

45.  Groups  belonging  to  the  different  values  of  a  function. 

46-47.  Transformation. 

48-50.  The  Cauchy-Sylow  Theorem. 

51 .  Distribution  of  the  elements  in  the  cycles  of  a  group. 

52.  Substitutions  which  belong  to  all  values  of  a  function. 

53.  Equation  for  a  p-valued  function. 

55.  Discriminants  of  the  functions  of  a  group. 

56-59.  Multiple- valued   functions,  powers  of   which    are    single- 

valued. 

CHAPTEE  ly. 

TRANSITIVITY    AND    PRIMITIVITY. SIMPLE    AND    COMPOUND    GROUPS. 

ISOMORPHISM. 

60-61 .  Simple  transitivity. 

62-63.  Multiple  transitivity. 

64 .  Primitivity  and  non-primitivity. 

65-67 .  Kon-primitive  groups. 

68.  Transitive  properties  of  groups. 

69-71.  Commutative  substitutions;  self -con  jugate  subgroups. 

72-73.  Isomorphism. 

74-76.  Substitutions  which  affect  all  the  elements. 

77-80.  Limits  of  transitivity. 

81-85.  Transitivity  of  primitive  groups. 

86.  Quotient  groups. 

87.  Series  of  composition. 

88-89.  Constant  character  of  the  factors  of  composition. 

91 .  Construction  of  compound  groups. 

92.  The  alternating  group  is  simple. 

93 .  Groups  of  order  p'^. 

94.  Principal  series  of  composition. 

95.  The  factors  of  composition  equal  prime  numbers. 

96 .  Isomorphism . 

97-98.  The  degree  and  order  equal. 

99-101 .  Construction  of  isomorphic  groups. 

CHAPTER  V. 

ALGEBRAIC     RELATIONS     BETWEEN     FUNCTIONS     BELONGING     TO     THE    SAME 
GROUP. FAMILIES    OF    MULTIPLE-VALUED    FUNCTIONS. 

103-105.        Eunctions  belonging  to  the  same  group  can  be  rationally 
expressed  one  in  terms  of  another. 

106,  Eamilies;  conjugate  families. 

107.  Subordinate  families. 


CONTENTS.  IX 

108-109.        Expression  of   the    principal    functions  in  terms  of   the 
subordinate. 

110.  The  resulting  equation  binomial. 

111 .  Functions  of  the  family  with  non- vanishing  discriminant. 

CHAPTER  VI. 

THE    NUMBER    OF    THE    VALUES    OF    INTEGRAL    FUNCTIONS. 

112.  Special  cases. 

113.  Change  in  the  form  of  the  question. 

114-115 .  Functions  whose  number  of  values  is  less  than  their  degree. 

116.  Intransitive  and  non-primitive  groups. 

117-121.  Groups  with  substitutions  of  four  elements. 

122-127 .  General  theorem  of  C.  Jordan. 

CHAPTER  VII. 

CERTAIN    SPECIAL    CLASSES    OF    GROUPS. 

128.  Preliminary  theorem. 

129.  Groups  i2  with  r  =  w  — p.    Cyclical  groups. 

130.  Groups  i2  with  r  =  n=p,  q. 

131.  Groups  Q  with  r  =  n=p'^. 

132-135.        Groups  which  leave,  at  the  most,  one  element  unchanged.— 

Metacyclic  and  semi-metacyclic  groups. 
136.  Linear  fractional  substitutions.     Group  of   the  modular 

equations. 
137-139.        Groups  of  commutative  substitutions. 

CHAPTER  VIII. 

ANALYTICAL     REPRESENTATION    OF    SUBSTITUTIONS. THE     LINEAR     GROUP. 

140.  The  analytical  representation. 

141 .  Condition  for  the  detining  function. 

143.  Arithmetic  substitutions. 

144.  Geometric  substitutions. 

145.  Condition  among  the  constants  of  a  geometric  substitution. 
146-147.  Order  of  the  linear  group. 


PART  n. 

Application  of  the  Theory  of  Substitutions  to  the  Algebraic 

Equations. 

CHAPTER  IX. 

the   equations  of   the  second,  third  and  fourth  degrees. — group 

of  an  equation. — resolvents. 
148.  The  equations  of  the  second  degree. 


X  CONTENTS. 

149 .  The  equations  of  the  third  degree. 

150.  The  equations  of  the  fourth  degree. 

152 .  The  general  problem  formulated.    Galois  resolvents. 

153-154 .  Affect  equations.    Group  of  an  equation. 

156.  Fundamental  theorems  on  the  group  of  an  equation. 

157 .  Group  of  the  Galois  resolvent  equation. 
158-159.  General  resolvents. 

CHAPTER  X. 

THE    CYCLOTOMIC    EQUATIONS. 

161.  Definition  and  irreducibility. 

162 .  Solution  of  cyclic  equations. 

163.  Investigation  of  the  operations  involved. 
164-165.  Special  resolvents. 

166 .  Construction  of  regular  polygons  by  ruler  and  compass. 

167.  The  regular  pentagon. 

168 .  The  regular  heptadecagon. 

169-170 ,        Decomposition  of  the  cyclic  polynomial. 

,    CHAPTER  XI. 

THE    ABELIAN    EQUATIONS. 

171-172.  One  root  of  an  equation  a  rational  function  of  another. 

173.  Construction  of  a  resolvent. 

174-175.  Solution  of  the  simplest  Abelian  equations.     ' 

176 .  Employment  of  special  resolvents  for  the  solution. 

177 .  Second  method  of  solution. 
178-180.  Examples. 

181.  Abelian  equations.    Their  solvability. 

182.  Their  group. 

183.  Solution  of  the  Abelian  equations;  first  method. 
184-186.  Second  method. 

187 .  Analytical  representation  of  the  groups  of  primitive  Abelian 

equations. 
188-189.        Examples. 

CHAPTER  XII. 

EQUATIONS  WITH  RATIONAL  RELATIONS  BETWEEN  THREE  ROOTS. 

190-193 .        Groups  analogous  to  the  Abelian  groups. 
194.  Equations  all  the  roots  of  which  are  rational  functions  of 

two  among  them. 

196.  Their  group  in  the  case  n=p. 

197 .  The  binomial  equations. 


CONTENTS. 


XI 


199.  Triad  equations. 

200-201 .  Constructions  of  compound  triad  equations. 

202 .  Group  of  the  triad  equation  for  n  =  l. 

203-205.  Group  of  the  triad  equation  for  7i  =  9 

206 .  Hessian  equation  of  the  ninth  degree. 

CHAPTER  XIII. 

THE    ALGEBRAIC    SOLUTION    OF    EQUATIONS. 

207-209.  Rational  domain.    Algebraic  functions. 

210-211.  Preliminary  theorem. 

212-216.  Roots  of  solvable  equations. 

217.  Impossibility  of  the  solution  of  general  equations  of  higher 

degrees. 

218.  Representation  of  the  roots  of  a  solvable  equation. 

219.  The  equation  which  is  satisfied  by  any  algebraic  expression. 
220-221 .        Changes  of  the  roots  of  unity  which  occur  in  the  expres- 
sions for  the  roots. 

222-224 .        Solvable  equations  of  prime  degree. 

CHAPTER  XIV. 

THE    GEOUP    OF    AN    ALGEBRAIC    EQUATION. 

226.  Definition  of  the  group. 

227 .  Its  transitivity. 

228.  Its  primitivity. 

229.  Galois  resolvents  of  general  and  special  equations. 

230.  Composition  of  the  group. 

231 .  Resolvents. 

232-234.        Reduction  of  the  solution  of  a  compound  equation. 
235 .  Decomposition  of  the  equation  into  rational  factors. 

236-238.        Adjunction  of  the  roots  of  a  second  equation. 

CHAPTER  XY. 

ALGEBRAICALLY    SOLVABLE    EQUATIONS. 

239-241 .  Criteria  for  solvability. 

242 .  Applications. 

243 .  Abel's  theorem  on  the  decomposition  of  solvable  equations. 

244.  Equations  of  degree  p^;  their  group. 

245.  Solvable  equations  of  degree  _p. 

246.  Solvable  equations  of  degree  _p^ 

248-249 .        Expression  of  all  the  roots  in  terms  of  a  certain  number  of 
them. 


EKKATA. 

p.      7,  footnote,  for  transformatione  read  transmutatione. 

p.  15,  line  10,  read  (p  =  S  s/~^ . 

p.  16,  line  5,  read  s^i  —  tp^  —  ^S^s/ A. 

p.  28,  line  9,  for  cp  read  ^. 

p.  29,  line  12,  for  ^read  ^. 

p.  29,  line  9,  from  bottom,  for  4^  read  ^. 

p.  31,  line  8,  read  G  —  [1,  {x^xi)  {x^x^,  {xxX^  {x^x^,  ix^x^  {^i^^^ 

p.  41,  line  7,  for  p  read  p-^. 

p.  52,  line  13,  read  aip^-\-\)il,\. 

p.    52,  line  5,  from  bottom,  for  -—  read  — . 

da.  Cia 

p.    89,  line  2,  for  not  more  read  less. 

p.    93,  line  9,  for  a  group  H  read  a  primitive  group  H. 

p.    94,  line  2,  for  n — q-\-2  read  n  —  q-\-k. 

p.    98,  line  19,  for  §J',§,  read  §«§^(t,. 

p.  101,  line  3,  for  ^2,  read  Q\. 

p.  103,  line  14,  read  [1,  (2:12:2)]. 

p.  125,  Theorem  XI,  read:    In  order  that  there  may  be  a pp-valued 

function  /   a  prime  power  /f   of  which  shall   have   p 

values,  etc. 
p.  159,  lines  10,  11  from  bottom,  read:    Since  <Tj  belongs  to  ?'i ,  at 

least    one  of    the  exponents    p.,  v ,  .  .  .  ,    the  s  of  which 

belongs  to  r^,  must  be  prime  to  ?'i. 
p.  166,  line  4,  read  r^ ,  /2'2  >  ^3^2 »  •  •  • 
p.  174,  line  2,  for  Cj  read  2ci . 
p.  210,  foot  note,  for  No.  XI,  etc.,  read  478-507,  edition  of  Sylow 

and  Lie. 
p.  219,  line  10,  for  ^{cos  a)  read  ffi(cos  a). 
p.  224,  line  2  from  bottom,  read:  which  leaves  two  elements  with 

successive  indices  unchanged. 
p.  248,  line  3  from  bottom,  read  V^%V  =  ^a  +  ^(Va+2,  • .  •)• 


PART  I. 

THEOKY  OF    SUBSTITUTIONS  AND  OF   THE    INTEGKAL 

FUNCTIONS. 


CHAPTER  I. 

SYMMETRIC    OR    SINQLE  -  VALUED    FUNCTIONS.    ALTERNA- 
TING AND  TWO-VALUED  FUNCTIONS. 

§  1.     In  the  present  investigations  we  have  to  deal  with  n  ele- 
ments ^1,  X.,  .  .  .  x„,  which  are  to  be  regarded  throughout  as  entirely- 
independent  quantities,  unless  the  contrary  is  expressly  stated.     It 
is  easy  to  construct  integral  functions  of  these  elements  which  are 
unchanged  in  form  when  the  x^^s  are  permuted  or  interchanged  in. 
anyway.     For  example  the  following  functions  are  of  this  kind: 
^i"  +  0-2*  +  a?/  -f  . . .  +  x^", 
x^'^  x./  -\-  Xi'^  x./  -\-  . .  .  -{-  x.^  0?!^  -1-  a?/  ^^3^  -f-  ... 
+  X,''  Xi^  ...  +  x,,"  x^,,_i, 

[x^  —  x,)^  (xi — x^y  {x.2  —  x^Y  . . .  (x„_i  —  x„y, 

etc. 

Such  functions  are  called  symmetric  functions.  We  conjQne 
ourselves,  unless  otherwise  noted,  to  the  case  of  integral  functions. 

If  the  Xx's  be  put  equal  to  any  arbitrary  quantities,  a^ ,  a2)  •  •  -(^m 
?o  that  Xj  =  ttj,  Xo  =  a.y,  ...  x„  =  a„,  it  is  clear  that  the  symmetric 
functions  of  the  x^'s  will  be  unchanged  not  only  in  form,  but  also  in 
value  by  any  change  in  the  order  of  assignment  of  the  values  a^,  to 
the  X\S.     Such  a  reassignment  may  be  denoted  by 

X  =  tti  ,  Xo  =  cii ,  ...  Xn  =  a, 
where  the  a,^ ,  a,-^ ,  .  .  .  denote  the  same  quantities  a^,  a2,  ...  in  any 
one  of  the  possible  n !  orders. 

Conversely,  it  can  be  shown  that  every  integral  function, 
^(Xj,  X2  .  .  .   x„)  ,  of  n  independent  quantities  x^,  x^,  ...  x,,,  which 


THEORY    OF    SUBSTITUTIONS. 


is  unchanged  in  value  by  all  the  possible  permutations  of  arbitrary- 
values  of  the  X\S,  is  also  unchanged  in  form  by  these  permutations. 

Theorem  I.  Every  single-valued  integral  function  of  n 
independent  elements  x^,  a?2,  ...  x,^  is  symmetric  in  these  elements. 

§  2.  The  reasoning  on  which  the  proof  of  this  theorem  is 
based  will  be  of  frequent  application  in  the  following  treatment. 
It  seems,  therefore,  desirable  to  present  it  here  in  full  detail. 

(A).     If  in  the  integral  function 


A=n 


<1) 


/  (.t)  =2     ttA  a;' 


/.  =  o 


all  the  coefficients  a^  are  equal  to  zero,  then  /  (x)  vanishes  identi- 
cally, i.  e.,  f(x)  is  equal  to  zero  for  every  value  of  x.  Conversely, 
if  f{x)  vanishes  for  every  value  of  x,  then  all  the  coefficients  a\  are 
«qual  to  zero. 

For  if  f{x)  is  not  identically  zero,  then  there  is  a  value  C^  guch 
that  for  every  real  x  of  which  the  absolute  value  j  .t  |  is  greater  than 
Co ,  the  value  of  the  function  f{x)  is  different  from  zero.  For  Co  we 
may  take  the  highest  of  the  absolute  values  of  the  several  roots  of 
the  equation  f{x)  =  0.  Without  assuming  the  existence  of  roots 
of  algebraic  equations,  we  may  also  obtain  a  value  of  Co  as  follows:* 

Let  %  be  the  numerically  greatest  of  the  n  coefficients  Go,  cii, 


.  .  a„_i  in  (1),  and  denote 


a„\ 


by  r.     We  have  then 


Ctn- 


X" 


+  .  .  .   +  f/o 


< 


(h 


< 


^+i.^r^+...-M) 


(r-l) 


Hence,  for  any  value  of  x  not  lying  between 


—1 

r  and  +  r, 

<  i^r 

SO  that  the  sign  of  f{x)  is  the  same  as  that  of  a„  x".     Consequently, 
we  may  take  Co  =  ^*. 


a,, 

_,x"- 

^  +  a„_2a:"-^+  . 

..+a. 

a,, 

*L.  Kronecker.   Crelle  101,  p.  347. 


SYMMETRIC    AND    TWO-VALUED    FUNCTIONS.  6 

(B).     If  no  two  of  the  integral  functions 

(2)  ./,  (4 /.(*),... /.,(*) 

are  identically  equal  to  each  other,  then  there  is  always  a  quantity 
Xo  such  that  for  every  x  the  absolute  value  of  which  is  greater  than 
Co,  the  values  of  the  functions  (2)  are  different  from  each  other. 

For,  if  we  denote  by  i\p  the  value  determined  for  the  function 
fa  {^)  — //3  {x)i  as  r  was  determined  in  (A),  we  may  take  for  Co  the 
greatest  of  the  quantities  Va^. 

(C).     If  in  the  integral  function 

/(.rj,  X,,    .  .  .   X,)    =  S  «Al  X2.  .  .  \>n  Xi^^  x^2  .  .  .  xj>» 

all  the  coefficients  a  are  equal  to  zero,  then  the  function  /  vanishes 
identically,  i.  e.,  the  value  of  /  is  equal  to  zero  for  every  system  of 
values  of  x^ ,  x., ,  . . .  x„. 

To  prove  the  converse  proposition  we  put 

(3)  x^,  =  g,     a:,  =  gv,     x,=:gi>\       ,  ,  .  x„  =  gi-"'-' 

a^i^^  .T>  .  .  .  .r,/'"  then  becomes  a  power  of  g,  the  exponent  of 
which  is 

^Ai  A2  .  .  .  A.-  =  ^1  +  ^-^  +  ^;)>'  +   .  .  .  Ky-' 

From  (B),  we  can  find  a  value  for  v  such  that  for  all  greater  val- 
ues of  V,  the  various  rxi  X2  •  •  •  Am  are  all  different  from  one  another. 
We  have  then 

/  (^1,  OC2,     ...    X,,)    =   2  aAlA2    .   .   .    \,n   fy'"^^  ^i'"    ^n 

But,  from  (A),  if  all  the  coefficients  a  do  not  disappear,  we  can 
take  g  so  large  that  /  is  different  from  0.  The  converse  proposition 
is  then  proved. 

(D).     If  a  product  of  integral  functions 

(4)  /i  (^1  ,    ^2  )    •   •   •   '^n)    J  2  (^1  )  ^2  )    •   •   •    X„)    .   .   .  f^i  {Xi  ,  X2,    .  .   .    X,^) 

is  equal  to  zero  for  all  systems  of  values  of  the  x^^s,  then  one  of  the 
factors  is  identically  zero. 

For,  if  we  employ  again  the  substitution  (3),  we  can,  from  (C), 
select  such  values  ga  and  v/^for  any  factor /«  {x^,  x^,  ...  ir„)  which 
does  not  vanish  identically,  that  for  every  system  of  values  which 
arises  from  (3 )  when  g  >  ga  and  >  >  v^  the  value  of  /«  is  different 
from  zero.  If  then  we  take  g  greater  than  gx,  gii  -  -  -  Q,n  and  at  the 
same  time  >  greater  than  v,^ ,  ^2 ,  ...  ^,n  we  obtain  systems  of  values 


4  THEORY    OF    SUBSTITUTIONS. 

of  the  Xxs  for  which  (4)  does  not  vanish,  unless  one  of  the  factors 
vanishes  identically. 

The  proof  of  Theorem  I  follows  now  directly  from  (C).  For  if 
<p  (x^,  Xo,  .  .  .  x„)is  a  single- valued  function,  and  if  c^i  (x^,  x.2  .  .  .  x„) 
arises  from  <p  by  any  rearrangement  of  the  X\s,  then  it  follows  from 
the  fact  that  <p  has  only  one  value,  that  the  difference 

(p  {x,,  Xo^,  .  .  .  X,,)  —^1  {x,,X2,  .  .  .  X,,) 

vanishes  identically. 

If  the  elements  X\  are  not  independent,  Theorem  I  is  no  longer 
necessarily  true.  For  instance,  if  all  the  a-^'s  are  equal,  then  any 
arbitrary  function  of  the  x^s  is  single- valued.  Again  the  function 
Xi^  +  dx^~  X2 — 4:Xi  X2  +  3  X2  is  single- valued  if  x^^  =  2x.2^  although 
it  is  unsymmetric. 

§  3.  If,  in  any  symmetric  function,  we  combine  all  terms  which 
only  differ  in  their  coefficients  into  a  single  term,  and  consider  any 
one  of  these  terms,  Cx^"-  x.^  x^^  .  .  . ,  then  the  symmetric  character 
of  the  function  requires  that  it  should  contain  every  term  which  can 
be  produced  from  the  one  considered  by  any  rearrangement  of  the 
X\^.  If  these  terms  do  not  exhaust  all  those  present  in  the  func- 
tion there  will  still  be  some  term,  Cx^^'  x.f'  x-^' ...  in  which  the  sys- 
tem of  exponents  is  not  the  same  as  in  the  preceding  case.  This 
term  then  gives  rise  to  a  new  series  of  terms,  and  so  on.  Every 
symmetric  function  is  therefore  reducible  to  a  sum  of  simpler  sym- 
metric functions  in  each  of  which  all  the  terms  proceed  from  any 
single  one  among  them  by  rearrangement  of  the  Xx^.  The  several 
terms  of  any  one  of  these  simple  functions  are  said  to  be  of  the 
same  type  or  similar.  Since  these  functions  are  deducible  from  a 
single  term,  it  will  suffice  to  write  this  one  term  preceded  by  an  S. 
Thus  S  (x^)  denotes  in  the  case  of  two  elements  Xi"  +  x.2^,  in  the 
case  of  three  Xi^  -\-  x}  +  x-^^  etc. 

§  4.  If  we  regard  the  elements  x^^  x<2^  ...  x,,  as  the  roots  of  an 
equation  of  the  n^^  degree,  this  equation,  apart  from  a  constant  fac- 
tor, has  the  form 

(5)  /"(ic)  =  {x  —  x^  (a?  —  .T2)  . .  .  {^x  —  a?,,)  =  0 

the  left  member  of  which  expanded  becomes 


SYMMETRIC    AND    TWO-VALUED    FUNCTIONS.  O 

(6)  a?"  —  (a?i  +  0-2  +  . . .  +  ^«)  ^''~' 

-\-  (^Xi  X2  ~T~  Xi  X-j  -\-  X2  X^  -}-   .  .  .  -J-  Xn_i  X„)X 
•  •  I    ~|~  (         J-  )    X-^  X2   t  •  .   X,f • 

=  a?"  —  Ci  a?""^  +  Cacc""^ —  •  •  •  +  ( — l)"c» 
The  coefficients  of  the  powers  of  x  in  this  equation  are  there- 
fore simple  integral  symmetric  functions  of  the  x^s: 

(7)  01  =  8(^-1),       02  =  8(071X2)5      Cx  =  S  (a-i  ^2  . . .  a^x), 

•   •   •   •   •  Cjj   O  \,«^i  «^2    •   •   •   *^n)    '^1  *^2   •   •   •    *^n 

These  combinations  Ca  are  called  the  elementary  symmetric 
functions.  They  are  of  special  importance  for  the  reason  that  every 
symmetric  function  of  the  x^s  can  be  expressed  as  a  rational  inte- 
gral function  of  the  c^'s. 

§  5.  Among  the  many  proofs  of  this  proposition  we  select  that 
of  Gauss.  * 

We  call  a  term  a^i^'i  a^2"'2  x^''^ .  .  .  higher  than  x^^^  x./-"^  a^g^s  .  .  .  when 
the  first  of  the  differences  m^  —  /^i ,  ma  —  p.2 ,  ni^  —  !H  ?  •  •  •  which 
does  not  vanish  is  positive.  This  amounts  then  to  assigning  an  arbi- 
trary standard  order  of  precedence  to  the  elements  X)^. 

In  accordance  with  this  convention,  Cj,  C2,  Cg,  . . .  c^,  .  .  .  have 
for  their  highest  terms  respectively 

ry*  ^y*     'y*  /y     ^y*     ^y*  ^     ^y*     ^y>  ^y*. 

1    ?  tAyi     <A^2    9  «A>1     t4^9    *^  3   9  •     •     •      tA^I     «A-9    *^3      •     •     •      '^'A  5  •     •     • 

and  the  function  Ci*  Ca^  c^"^  -  .  .  has  for  its  highest  term 

^^a  +  ^  +  V  +  . . .  ^2^  +  V  +  . . .  a^gV  +  • . ._ 

In  order,  therefore,  that  the  highest  terms  of  the  two  expressions, 
Ci*  Co^  Cg'y  .  .  .  and  c^^-'  c^^'  c-^'  .  .  .  may  be  equal,  we  must  have 

/5  +  r+  ...  =         /?'+/+  ... 
r  +  . . .  =  r'  +  . . . 


that  is,  a  =  a',  /?  =  y9',  ^  =  ^',  .  .  . 

It  follows  that  two  different  systems  of  exponents  in  c^  c.,^  c-i' .  .  , 
give  two  different  highest  terms  in  the  Xv's.     Again  it  is  clear  that 

x^^X2^x;i  .,.  (  «  i  /5  >  r  =  ^ . . .) 

is  the  highest  term  of  the  expression  c^°-~^  c<^~ '^  c^~^ .  .  .  and  that 

*  Deinonstratio  nova  altera  etc.    Gesammelte  Werke  III,  §  5,  pp.  37-38.   Cf.  Krou- 
ecker,  Monatsberichte  der  Berliner  Akademie,  1889,  p.  943  seq. 


6  THEORY    OF    SUBSTITUTIONS. 

all  the  terms  in  the  expansion  of  this  expression  in  terms  of  the  e^^'s 
are  of  the  same  degree. 

§  6.  If  now  a  symmetric  function  S  be  given  of  which  the 
highest  term  is 

A  a-i*^  x/x^y  x^^  .  .  .  (  «  =  /5  >  ^  >  o\  .  . ) 

the  difPerence 

S—  A  c,--^  c/-v  c,y-^  ...  =S, 

will  again  be  a  symmetric  function ;  and  if,  in  the  subtrahend  on  the 
left,  the  values  of  the  Ca.'s  given  in  (7)  be  substituted,  the  highest 
term  of  S  will  be  removed,  and  accordingly  a  reduction  will  have 
been  effected. 

If  the  highest  term  of  S^  is  now  A^  x{^'  x.f'  x^y'  x^^'  .  .  . ,  then 

s.  —  A.c.'^'-^'cf-y'c/-^'...  =  s, 

is  again  a  symmetric  function  with  a  still  lower  highest  term.  The 
degrees  of  S.2  and  S^  are  clearly  not  greater  than  that  of  S,  and  since 
there  is  only  a  finite  number  of  expressions  Xi^  x^}^  Cfg^^  .  .  .  of  a  given 
degree  which  are  lower  than  x^°-  x.f  x^y  .  .  . ,  we  shall  finally  arrive  by 
repetition  of  the  same  process  at  the  symmetric  function  0 ;  that  is 

S,  -  A,  c.«<^>  -  ?«  c/''->'-  vW  .  .  .  =  0 ; 
and  accordingly  we  have 

S  =  A^  Ci--^c/-Y  ...  +  A2Ci«'-^'c/'-v'...  +  ... 

§  7.  It  is  also  readily  shown  that  the  expression  of  a  symmetric 
function  of  the  cca's  as  a  rational  function  of  the  Cx's  can  be  effected 
in  only  one  way. 

For,  if  an  integral  symmetric  function  of  x-^,  x^-,  ...  x^  could 
be  reduced  to  two  essentially  different  functions  of  c-^,  c^,  ...  c„, 
9?  (Cj,  C2,  . . .  c„)  and  v''  (ci,  C2,. .  .  c„),  then  we  should  have,  for  all 
values  of  the  a-^'s,  the  equation 

<P  (ci,  C2,  . . .  c„)  =  4'  (Ci,C2,  ...  c,.) 

The  difference  <p  —  (/',  which,  as  function  of  the  Ca's,  is  not 
identically  zero,  since  otherwise  the  two  functions  <p  and  ^  would 
coincide,  must,  as  function  of  the  x^^s,  be  identically  zero. 

Suppose,  now,  that  in  <p  —  v''  those  terms  in  Ci ,  C2 ,  ...  c„  which 
cancel  each   other    are  removed,  and    let  any  remaining  term   be 


SYMMETRIC    AND    TWO-VALUED    FUNCTIONS.  i 

B  Ci*  Co^  Cg"*'  .  .  .      This  term  on  being  expressed  in  terms  of  the  a^^'s 
will  give  as  highest  term 

Now  the  different  remaining  terms  5'  cf  c.f  c/ .  .  .give  different 
highest  terms  in  the  X\S  (§5).  Consequently  among  these  highest 
terms  there  must  be  one  higher  than  the  others.  But  the  coefficient 
of  this  term  is  not  zero;  and  consequently  (§2  (C))  the  function 
(f  —  </'  cannot  be  identically  zero.     We  have  therefore 

Theorem  II.  An  integral  symmetric  function  of  x-^,  X2,...  Xn 
can  always  be  expressed  in  one  and  only  one  way  as  an  integral 
funGtio7i  of  the  elementary  symmetric  functions  c^,  Cg,  ...  c„. 

§  8.  If  we  write  Sa  =  S  (a?/)  for  the  sum  of  the  X  powers  of 
the  n  elements  Xi,  Xo,  .  . .  Xn,  we  might  attempt  to  calculate  the  Sa's 
as  functions  of  the  Ci's  by  the  above  method.  It  is  however  simpler 
to  obtain  this  result  by  the  aid  of  two  recursion  formulas  first  given 
by  Newton  *  and  known  under  his  name.     These  formulas  are 

A)  s,— Ci s,_i  +  C2S._2—  . . .  +  (—1 )" c,,s,_„  =  0      (  ^  >  ^  ) 

B)  s,  —  CiS,._i  -{-c,s,._2 —  .  .  .  +  (  — 1)^  r  c,.—  0        {r  ^n) 
These  two  formulas  can  be  proved  in  a  variety  of  ways.     The 

formula  A)  is  obtained  by  multiplying  the  right  member  of  (6) 
by  x^^^,  replacing  a?by  a?x,  and  taking  the  sum  over  /  =  1,  2,  .  .  .  w . 
The  formula  B )  may  be  verified  with  equal  ease  as  follows.  If 
we  represent  the  elementary  symmetric  functions  oi  X2,  x^,  .  .  .  x^ 
by  c/,  Co',  .  .  .  c'„_i,  we  have 

^1  =  »^1  -f"  Cj  J         C2  ^  Xi  Ci    ~t~  C2  5       C3  :=  CC]  C2   -p  C3  ,  .  .  . 

and  accordingly,  if  r  ^  n,  we  have 

x/  —  c,  x/^'  +  C2 o^/-^—  .  .  .  (— l)^c,      . 
=  ^1^— (^1  +  c/)  ^1^-^  +  (a^c/  +C2')^/- — .  .  . 
+  (-1/  {x,c\._,-^  cj)  =  (-l/c/ 
and  hence,  replacing  x^  successively  hj  x^,  x^,  .  .  .  x^,  and,  corres- 
pondingly, c,/  by  c/',  c/",  .  .  .  c/"\  and  taking  the  sum  of  the  n 
resulting  equations 

s,.  —  CiS,._i  +  C2S,._2  —  .  .  .  ( — lyc,.  n 

=  (  — 1  )r  (c/  +  cj'  4-  c,!"  +  .  :  .  +  c/"  ). 

*  Newton:    Arith.  Univ.,  De  Transformatione  Aequationum. 


O  THEORY    OF    SUBSTITUTIONS. 

The  right  member  is  symmetric  in  cci ,  X2,.  .  .  x„,  and  contains  all 
the  terms  of  c,.  and  no  others.  Moreover,  the  term  x^  x.,  .  .  .  x,. ,  and 
consequently  every  term,  occurs  71  —  7'  times.    Accordingly  we  have 

s,  — CiS,_i  +  CoS,_o—  .  .  .  -i-  {—ly  c,n  =  {—ly  {n—r)c,, 
.'.  s,.  —  Cj  s,. _  1  +  C2  s,. _2  —  ...  ( —  1 )''  c,. r  =  0, 
and  formula  B)  is  proved.  *     The  formula  A)  can  obviously  be  veri- 
fied in  the  same  way. 

§  9.     The  solution  of  the  equations  A)  and  B)  ior  the  successive 
values  of  the  s^s  gives  the  expressions  for  these  quantities  in  terms 
of  the  Cx's.      The  solution  is  readily  accomplished  by  the   aid  of 
determinants.     We  add  here  a  few  of  the  results. f 
C)     So  =  n 

02    Cj  ^^2 

S3    =    Ci^  ' dCiC2   +    SCg 

54  =  Cj^  —  4ci^C2  +  4C1C3  -\-  2c2^  —  4iC^     ■ 

55  =  Ci^  —  5ci^C2  +  5ci%  +  ^CiCo^  —  ^c^c^  —  5C2C3  +  5C5 

It  is  to  be  observed  here  that  all  the  c^'s  of  which  the  indices  are 
greater  than  n  are  to  be  taken  equal  to  0.  This  is  obvious  if  we  add 
to  the  n  elements  x^,  x^,  ...  x,^  any  number  of  others  with  the 
value  0;  for  the  c^'s  up  to  c„  will  not  be  affected  by  this  addition, 
while  c„+i,  c„+2,  ...  will  be  0. 

§  10,  The  observation  of  §  5  that  Cj*  Cg^  c^y .  .  .  gives  for  its  highest 
term  a^i*  +  ^  +  >'  +  •••  a?2^  +  "^  +  •••  x^y,-^,--;  can  be  employed  to  facili- 
tate the  calculation  of  a  symmetric  function  in  terms  of  the  C\S. 

We  may  suppose  that  the  several  terms  of  the  given  function 
are  of  the  same  type,  that  is  that  they  arise  from  a  single  term 
among  them  by  interchanges  of  the  X\S .  The  function  is  then 
homogeneous;  suppose  it  to  be  of  degree  j>.  We  can  then  obtain 
its  literal  part  at  once. 

For,  if  the  function  contains  one  element,  and  consequently  all 
elements,  in  the  m^^  and  no  higher  power,  then  every  term  of  the 
corresponding  expression  in  the  c^'s  will  be  of  degree  m  at  the  high- 
est.    For,  in  the  first  place,  two  different  terms  Ci<^  c/  c^y  .  .  .  and 

♦Another,  purely  arithmetical,  proof  is  given  by  Euler;  Opuscula  Varii  Argumenti. 
Demonstrat.  genuina  theor.  N«ewtoniani,  II,  p.  108. 
+  C/.  FaA  di  Bruno:  Formes  Binalres. 


SYMMETRIC    AND    TWO-VALUED    FUNCTIONS.  9 

Cj*'  c.f  c/  .  .  .  give  different  highest  terms  in  the  a^x's,  so  that  two 
such  terms  cannot  cancel  each  other;  and,  in  the  second  place, 
€j;^€.f  c^y  ...  gives  a  power  X}^'^  +  ^  +  y  +••-,  so  that 

«  +  A^  +  r  +  .  . .  i  m. 

Again  the  degree  of  x^""  -^  ^  +  y  +  •••  x.,^^  ^  -^  •••  x^y  -^  •.••.  .    is 

a  +  2  ,?  +  3  r  +  . . .  =  ^ 

and     since     the    given     expresssion    is     homogeneous,    the    sum 

u-\-2l^-\-dy-\-  ...  must  be  equal  to  v  for  every  term  c^^  c./  e^y  .  .  . 

These  two  limitations  imposed  on  the  exponents  of  the  c^'s  that 

«  +  /5  +  r  +  .  .  .  £  ^n,  a  +  2  ,5  +  3  r  .  .  .  =  ^ 

exclude  a  large  number  of  possible  terms.  The  coefficients  of  those 
that  remain  are  then  calculated  from  numerical  examples.  The 
quantity  a-^2l3-\-Sy-\-...  is  called  the  ivelght  of  the  term 
^1**  ^2^  CgV .  . .  and  a  function  of  the  c^^s  whose  several  terms  are  all  of 
the  same  weight  is  called  isobarlc.  For  example 
S  (xi^  X2^  xi  a?4 )  =  go^T  +  gi  Cg  c^  +  g.,  %  c,  +  q^  c^  c^  (m  =  2,  v  =  7) 
S  [Soc,  —  x,)\x,  —  x,)'(x^  —  Xiy]  ^^oCfi  +  giCsC,  -\-q2C^c, 

+  ^3  G,  (^x  +  ^4  C3'  +  gs  C3  C2  Ci  +  q^  C3  Ci'  4-  q,  c/  +  gs  c/  c{ 

(m  =  4,  V  =  6) 
where  the  g's  are  as  yet  undetermined  numerical  coefficients. 

In  the  second  example  we  will  calculate  the  g's  for  the  case  n  =  3, 
for  which  therefore  c^  =  %=Gi^  —  0.  It  is  obvious  that  for  different 
values  of  n  the  coefficient  g's  will  be  different.     Taking 

I.     a?i  =:  1,  X2  —  —  1,  x.^  =:  0,  we  have   c^  =  0,  Cg  =  —  1,  c^  —  O 

.••S  =  4  =  -3,;  •  3:  =  -4. 

II.     Xi  =  X2  =  1,  a?3  =  0,    .  Cj  =  2,  C2  =  1,  Cg  =  0 

.-.  S  =  0  =  — 4  +  4^3;  gs^l. 

III.     Xi  =  X2=  1,  x^  =  —  1,  Ci  =  0,  0.2  —  —  3,  C3  =  —  2 

.-.   S  ==0  =  4g, +4 -27;  g,  =  — 27. 

lY.     Xi  =  Xo  =  2,  x^  =  —  1,  Ci  =  3,  C2  =  0,  Cg  =  —  4 

.♦.  S  =  0  =  — 2716— 4-27g6;    gfi  =  — 4. 
V.     a?i  =^  a?2  =  a?3  =  1,  Cj  =  3,  C2  =  3,  Cg  =  1 

.-.  S  =  0  =  — 27  +  9 gg  — 135;  q,  ==18. 

+  18  Cg  C2  Gi  4C3  Ci^  4:C.2^  +   62^  Cjl 


10  THEOKY    OF     SUBSTITUTIONS. 

This  expression,  (x^  —  XoY  {  X2  —  x-^y^x^  —  x^Y  =  A,  is  called 
the  discriminant  of  the  quantities  x^,  x^,  x^.  The  characteristic 
property  of  this  discriminant  is  that  it  is  symmetric  and  that  its 
vanishing  is  the  sufficient  and  necessary  condition  that  at  least  two 
of  the  Xx^  are  equal. 

§  11.  In  general,  we  give  the  name  "discriminant  of  n  quanti- 
ties Xi,  a?2,  ...  a?„"to  the  symmetric  function  of  the  x^^  the  van- 
ishing of  which  is  the  sufficient  and  necessary  condition  for  the 
equality  of  at  least  two  of  the  Xx^  .  If  a  symmetric  function  S  of 
the  XxS  is  to  vanish  for  x^  =  x.2,  it  must  be  divisible  by  x^  —  a?2, 
and  consequently  by  every  difPerence  Xa  —  x^.     Suppose 

S  =  {Xi  —  X2)  Si. 

Now  S,  and  consequently  (x^  —  x.2)  S^ ,  is  unchanged  if  x^  and 
X2  be  interchanged.  But  this  changes  the  sign  of  x^ — x.,  and  there- 
fore of  S^ .  Consequently  S^  vanishes  if  x^  =  X2,  and  accordingly 
S^  contains  x^  —  a?2  as  a  factor. 

The  symmetric  function  S  is  therefore  divisible  by  (x^  —  Xo)^ 
and  consequently  by  every  (Xa  —  ^^Y',  that  is  it  is  divisible  by 

J  =  \\  (xx  —  x^Y         {^^  <  /^-;  ^  =1,2,  ...  n  —  1;  ,a  =  2,  3,  .  .  .  71) 

AU. 

~~    V^^l  '^2)     \'^1  ^^3)     \'^1  "^ij    •  •   •    V*^!  '^nj 

(8)  (a?2  —  Xsf  (X2  —  x.f. ..  (X2  —  x„f 


This  quantity  A  already  satisfies  the  condition  as  to  the  equality 
of  the  XxQ ,  and,  being  the  simplest  function  with  this  property,  is 
itself  the  discriminant.  It  contains  ^n{n  —  1)  factors  of  the  form 
(xx—Xf^)"^',  its  degree  is  n{n  —  1),  and  the  highest  power  to  which 
any  Xx  occurs  is  the  (ii  —  1)^^.  It  is  the  square  of  an  integral, 
but,  as  we  shall  presently  show,  unsymmetric  function,  with  which 
we  shall  hereafter  frequently  have  to  deal. 

§  12.  Finally  we  will  consider  another  symmetric  function  in 
which  the  discriminant  occurs  as  a  factor. 

Let  the  equation  of  which  the  roots  are  x^,  x^,  . .  .x^  be,  as 
before,  f(x)  =  0.     Then  if  we  write 


SYMMETRIC    AND    TWO- VALUED    FUNCTIONS.  11 

we  have,  for  all  values  >^.  =  1,  2,  .  .  .  n,  the  equation 
f'{x^)  =  {xx  —  a?j)  (xx  —X2}... {xx  —  ^A_i) (a?A  —  a^x  +  1) .  .  . {x;,  —  x,,y 
We  attempt  now  to  express  the  integral  symmetric  function 

in  terms  of  the  coefficients  Cj ,  Co , .  .  .  c„  of  /  ( x),  Every  one  of  the 
71  terms  of  S  is  divisible  by  Xi  —  a'2,  since  either /'(xj)  or /'(;r2) 
occurs  in  every  term.  Consequently,  by  the  same  reasoning  as  in 
§  11,  S  is  divisible  by  {x^  — x._,y,  and  therefore  being  a  symmetric 
function,  by  every  {Xa  —  XpY,  that  is  by 

^  =TT(«^A  —  ^,^y        (^  <  /^-;    A  =  1,  2,  ...  71  —  1;  /.  =  2,  3,  .  .  .  ny 

S  is  therefore  divisible  by  the  discriminant  off{x),  i.  e.,  by  the  dis- 
criminant of  the  n  roots  of  f  (x). 

Now  f  {x\)  is  of  degree  of  n  —  1  in  Xa  and  of  degree  1  in  every 
other  Xfj,',  and  therefore 

^Y  •  f'i^-z)'  /'(a)  •  .  .  /'(^«)  is  of  degree  a  +  n  —  1  in  x^ 
x.2^  .  /'(^i)  .  f'{x^)  .  .  .  f'{x„)  is  of  degree        2n  —  3  in  a?i . 

Consequently,  if  a  <C  n  —  1,  >S  is  of  degree  2  n  —  3  in  Xj ,  while 
A  is  of  degree  2n  —  2  in  x^.  But  since  J  is  a  divisor  of  S,  it  fol- 
lows that  S  is  in  this  case  identically  0. 

(9)  S  [.r,».  /'(X3) .  fix,) .  .  .  f'(x,.)}  =  0,  («  <  n  -  1.) 

Again,  if  a  =  n  —  1,  then  S  and  J  can  only  differ  by  a  constant 
factor.  To  determine  this  factor  we  note  that  the  first  term  of  S  is 
of  degree  2  n  —  2  in  a^j ,  while  all  the  other  terms  are  of  lower 
degree  in  x^.     The  coefficient  of  x^^"^^  is  therefore 

y      J-)       {X2       x^) ,  .  .  {X2       Xn)  \x^       X2)  .  .  .  [x^       x,t) .  .  . 

n(n-l) 

{X,  —  Xo)  .  .  .  {X„  —  X„  _  1)  =  (—  1)       2         {X2  —  X^f  (X.2  —  X,y  .  .  . 

In  J  the  coefficient  of  a"i^"~^  is 

n(n  —  l) 
The  desired  numerical  factor  is  therefore  ( — 1)      2       and  wa 
have 


12  THEORY    OF    SUBSTITUTIONS. 

n(n-l) 

(10)     S[x-\f{x,),r{x,)...f'{x,^)-]  =  {-l)^-  J. 

Formulas  (9)  and  (10)  evidently  still  hold  if  we  replace  x^'^  or 

^i''~^  by  any  integral   function  c?  (x)  of  degree  a  ^  n  respectively. 
Moreover  since 

n(n—\) 

{-'^)~^~  ■^=nx,).f'{x,)...f'{x.,) 


■we  have 


p)  2^ 


^    =  0  or  1*, 


^a) 


according  as  the  degree  of  c  is  less  than  or  equal  to  n  —  1. 

§  13.  If  an  integral  function  of  the  elements  a*! ,  a:, ,  .  .  .  a^„  is 
not  symmetric,  it  will  be  changed  in  form,  and  consequently,  if 
the  Xxs  are  entirely  independent,  also  in  value,  by  some  of  the  possi- 
ble interchanges  of  the  a-^'s.  The  process  of  effecting  such  an  inter- 
change we  shall  call  a  suhstitution.  Any  order  of  arrangement  of 
the  Xx8  we  call  a  permutation.  The  substitutions  are  therefore 
operations;  the  permutations  the  result.  Any  substitution  whatever 
leaves  a  symmetric  function  unchanged  in  form ;  but  there  are  other 
functions  the  form  of  which  can  be  changed  by  substitutions.  For 
example,  the  functions 

I       Xi^ X2^  -}-  X^^ X^^,       Xi  X2   Xz  -\-  X^Xr^-\-  Xf,,       X^  -f"  X.2    +  '^'i 

take  new  values  if  certain  substitutions  be  applied  to  them;  thus  if 
iPj  and  X.J  be  interchanged,  these  functions  become 

JLJ.  tt/j     ~J~  X2     ~j~  X^  X^  ,         iCj    t4?2  «^3  '|~  '^Sj^o  ~\      "^6 )  2       1"       1     ~r"       3 ' 

The  first  two  functions  are  unchanged  if  x^  and  x^  be  inter- 
changed, the  second  also  if  x^  and  x^  be  interchanged,  etc. 

Functions  are  designated  as  one-  two-,  three-,  ??i-valued 
according  to  the  number  of  different  values  they  take  under  the 
operation  of  all  the  n!  possible  substitutions.  The  existence  of  one- 
valued  functions  was  apparent  at  the  outset.  We  enquire  now  as 
to  the  possibility  of  the  existence  of  two-valued  functions. 

In  §  11  we  have  met  with  the  symmetric  function  J,  the  dis- 
criminant of  the  n  quantities  cCi,  a'2,  ...  x„.  The  square  root  of  J 
is  also  a  rational  integral  function  of  these  n  quantities : 

*Tlie  formula  D  is  due  to  Euler;  Calc  Int.  II  §  116i?. 


SYMMETEIC    AND    TWO-VALUED    FUNCTIONS.  IS 

/y/J  =  (a?i  —  X2)  {Xi  —  0:3)  {xi  —  x^) .  .  .{xi  —  x„) 
(x^  --  x^)  (a?2  —  Xi).  .  .  {x,  —  Xn) 


\Xn  —  i  ^n)' 

Every  difference  of  two  elements  x^  —  x^  occurs  once  and  only 
once  on  the  right  side  of  this  equation.  Accordingly  if  we  inter- 
change the  aVs  in  any  way,  every  such  difference  still  occurs  once  and 
only  once,  and  the  only  possible  change  is  that  in  one  or  more  cases 
an  Xa  —  ^^  may  become  x^  —  Xa.  The  result  of  any  substitution 
is  therefore  either  +  V-J  or  —  \^  J,i.  e.,  the  function  V-^  is  either 
one-valued  or  two-valued.  But  if,  in  particular,  we  interchange  a?i 
and  X.2 ,  the  first  factor  of  the  first  row  above  changes  its  sign,  while 
the  other  factors  of  the  first  row  are  converted  into  the  correspond- 
ing factors  of  the  second  row,  and  vice  versa.  No  change  occurs  in 
the  other  rows,  since  these  do  not  contain  either  ^1  or  x., .  Since 
then,  for  this  substitution,  \/  J  becomes  —  \/  -^  >  it  appears  that  we 
have  in  V'-'  3-  two-valued  function. 

This  function  is  specially  characterized  by  the  fact  that  its  two 
values  only  differ  in  algebraic  sign.  Such  two- valued  functions  we 
shall  call  alternating  functions. 

Tlieoreiii  III.  The  square  root  of  the  discriminant  of  the  n 
quantities  x^,  x.j,  ...  Xn  is  an  alternating  function  of  these  quanti- 
ties. 

§  14.  Before  we  can  determine  all  the  alternating  functions,  a 
short  digression  will  be  necessary. 

xln  interchange  of  two  elements  we  shall  call  a  transposition. 
The  transposition  of  Xa  and  x^^  we  will  denote  by  the  symbol 
{Xa.  x^).     We  shall  now  prove  the  following 

Tlieoreni  IV.  Every  substitution  ca7i  be  replaced  by  a 
series  of  transpositions. 

Thus,  if  we  have  to  transform  the  order  x^,  X2,  x^,  .  .  .  x,^  into 
the  order    X[^,   Xi„,   x,:^,  .  .  .  xi,^,  we^apply  first   the   transposition 
(a'l  Xi^).     The  order  of  the  x^s  then  becomes  aj^^,  X2,  x^^  .  .  .  xi^  —  i 
iTj ,  Xi^  +  I,  ...  x„ ,  and  we   have   now   only   to   convert   the   order 

a'2  .  .  .  Xi^ _  1,  a-i ,  Xi^  +  I,  ...  x„  into  the  order  x^ y  xi^,  ,  .  .  xi^.      By 


14  THEORY    OF    SUBSTITUTIONS. 

repeating  the  same  process  as  before,  this  can  be  gradually  efPected, 
and  the  theorem  is  proved. 

Since  a  symmetric  function  is  unaltered  by  any  substitution,  we 
obtain  as  a  direct  result 

Theorem  V.  A  function  which  is  unchanged  by  every 
transposition  is  symmetric. 

§  15.  There  is  therefore  at  least  one  transposition  which 
•changes  the  value  of  any  alternating  function  into  the  opposite 
Talue.  We  will  denote  this  transposition  by  (Xa  oc^) ,  and  the  alter- 
nating function  by  c'-,  and  accordingly  we  have 

(,.'  yXi,  X^^  .  .   .   Xa  ;  •   •   •    X^  )  •   •   •    ^n)  ^'  V^^l  )    '^21    •   •   •    ^P^    •   •   •    '^aj    •   •   •    ♦^'«) 

Accordingly,  if  Xa  =  Xp,  we  must  have  v''  =  0.     Consequently  the 
-equation 

c''  (xi,  X2,  . .  .  z,  .  .  .  xp ,  .  . .  X,)  =  0 

regarded  as  an  equation  in  z  has  a  root  z  =  x^  and  the  polynomial 
(/'  is  therefore  divisible  by  z  —  x^.     The  function 

therefore  contains  Xa  —  cc^s  as  a  factor,  and,  consequently,  c''-  con- 
tains {Xa  —  Xp)-  as  a  factor. 

But  since,  for  all  substitutions,  c''  either  remains  unchanged  or 
only  changes  its  sign,  c'-^  must  be  a  symmetric  function;  and, 
accordingly,  since  </'-  contains  the  factor  (Xa  —  x^Y,  it  must  con- 
tain all  factors  of  the  form  (x^  —  x^)^,  i.  e.,  d'^  contains  J  as  a 
factor,  and  consequently  t'-  contains  V-J  as  a  factor.  The  remain- 
ing factor  of  (J>  is  determined  by  aid  of  the  following 

Theorem  VI.  Every  alternating  integral  function  is  of  the 
form  S.  \/  J  ,  where  \/  J  is  the  square  root  of  the  discriminant  and 
S  is^an  integral  symmetric  function. 

That  >S.  V'J  is  an  alternating  function  is  obvious.  Conversely, 
if  (}>  is  an  alternating  function,  it  is,  as  we  have  just  seen,  divisible 
by  V-J-  I^et  (V-j)'"  be  the  highest  power  of  V-J  which  occurs 
as  a  factor  in  c''.     Then  the  quotient 

is  either  a  one-  or  a  two-valued  function,  since  every  substitution 


SYMMETRIC    AND    TWO-VALUED    FUNCTIONS.  15 

either  leaves  both  numerator  and  denominator  unchanged  or  changes 
the  sign  of  one  or  both  of  them.  But  this  quotient  cannot  be  two- 
valued,  for  then  it  would  be  again  divisible  hj\/J,  which  is  con- 
trary to  hypothesis.  It  must  therefore  be  symmetric,  and  we  have 
accordingly 

<l>  =  s, .  (V  J)'" 

Now  if  m  were  an  even  number,  the  right  member  of  this  equa- 
tion, and  consequently  the  left,  would  be  symmetric.  We  must 
therefore  have  m  =  2n  -\-  1.     And  if  we  write  Si.  J"  =  S,  we  have 

^^  =  ^ .  V  J 

Corollary.  From  the  form  of  an  alternating  function  it 
follows  that  such  a  function  remains  uyichanged  or  is  changed  in 
sign  simultaneously  with  \/  J  for  all  substitutions. 

§  16.  Having  now  shown  how  to  form  all  the  alternating  func- 
tions, we  proceed  to  the  examination  of  the  two-valued  functions  in 
general. 

Let  <p{xi,  X2,  ...  x„)  be  any  two-valued  function,  and  let  the 
two  values  of  <p  be  denoted  by 

<fi (.Ti ,  X2,  .  .  .x„)     and     <p.2 (xi ,  x.2,  .  .  .  x,,). 

These  two  functions  must  differ  in  form  as  well  as  in  value,  and 
since  the  x^^s  are  any  arbitrary  quantities,  if  we  apply  to  <pi  and  ^2 
any  substitution  whatever,  the  resulting  values 

(«)  <Pi  (xi^ ,  xu,  ...  xu)     and     ^2  (^ii ,  Xu,  .  .  .  a?/„) 

will  also  be  different.  But  whatever  substitutions  are  applied  to  <p 
the  result  is  always  cr^  or  ^2  •  Consequently,  of  the  two  expressions 
(«),  one  must  be  identical  with  <pi  and  the  other  with  ^9  •  In  other 
words : 

Those  substitutions  which  leave  the  one  value  of  a  two-valued 
function  (f  unchanged,  leave  the  other  value  unchanged  also;  those 
substitutions  ichich  convert  the  one  value  of  f  into  the  other,  also 
convert  the  second  value  into  the  first. 

§  17.  From  the  preceding  section  it  follows  at  once  that  c-^  -\-  cr^ 
is  a  symmetric  function 

(y5)  <Pi  +  <f2  =  2Si. 


16  THEORY    OF    SUBSTITUTIONS. 

Again  the  difference  if^  —  cr2  is  a  two-valued  function  of  which 
the  second  value  is  ^2  —  s^i  =  —  (s^i  —  ^o)-  This  difference  is 
therefore  an  alternating  function,  and  accordingly,  from  Theorem 
VI,  we  may  write 

(r)  B,  —  s,  =  ls,s/~A. 

From  (/5)  and  (^)  we  obtain 

ci  =  ^1  +  s^  V^,    ^2  =  ^x  —  So  V^,    <p  =  s,  ±  s,  \/l. 

Conversely,  it  is  plain  that  every  function  of  the  last  form  is 
two-valued. 

Theorem  VII.  Every  two-valued  integral  function  is  of 
the  form  c^  =  Si  ±  S2  s/  J,  where  S^  and  S.2  are  integral  syniinetric 
functions  and  V  J  is  the  square  root  of  the  discriminant.  Con- 
versely every  function  of  this  form  is  two-valued. 

Corollary,  Every  two-valued  integral  function  is  unchanged 
or  changed  simultaneously  with  sj  Ahy  every  substitution. 

§  18.  From  the  corollaries  of  the  last  two  theorems  we  recog- 
nize the  importance  of  determining  those  substitutions  which  leave 
the  value  of  V  -^  unchanged. 

We  know  (§13)  that  the  transposition  {x^x-y)  changes  the  sign 
of  \/  J.  In  the  arrangement  of  the  factors  of  \/  J  in  the  same 
Section,  we  might  equally  well  have  placed  all  the  factors  containing 
Xa  or  Xp  in  the  first  and  second  rows,  Xa  —  x^  taking  the  place  of 
Xi  —  X2,  etc.  The  sign  of  \/  J  may  be  changed  by  this  rearrange- 
ment, but  whatever  this  sign  may  be,  it  will  be  changed  by  the 
transposition  (XaX^).  Consequently  \/  J  changes  its  sign  for  every 
transposition. 

This  result  is  easily  extended.  For,  if  we  apply  successively 
any  /j.  transpositions  to  V  J,  its  sign  will  be  changed  /j.  times,  that  is 
\/  J  becomes  ( —  l)'^  \/  J.  If  /j.  is  even,  V^  is  unchanged;  if  ,a  is 
odd,  V^  t)ecomes — \/ J.     We  have  therefore 

Theorem  VIII,  All  substituiions  which  are  formed  from 
an  odd  number  of  transpositions  change  the  value  ^  A  into  — V-J/ 
all  substitutions  which  are  formed  from  an  even  number  of  transpo- 
sitions leave  \/  J  unchanged.  Similar  results  hold  for  all  two-val- 
ued functions. 


SYMMETKIC    AND    TWO-VALUED    FUNCTIONS.  17 

§  19.  Every  substitution  can  be  reduced  to  a  series  of  transpo- 
sitions in  a  great  variety  of  ways,  as  is  readily  seen,  and  as  will  be 
shown  in  detail  in  the  following  Chapter.  But  from  the  preceding 
theorem  it  follows  that  the  number  of  transpositions  into  which  a 
substitution  is  resolvable  is  always  even,  or  always  odd,  according  as 
the  substitution  leaves  \/  J  unchanged  or  changes  its  sign. 

Theorem  IX.  If  a  given  substitution  reduces  in  one  ivay  to 
an  even  {odd)  number  of  transpositions,  it  reduces  in  every  way  to 
an  even  (odd)  number  of  transpositions. 

§  20.  Theorem  X.  Every  two-valued  function  is  the  root 
of  an  equation  of  the  second  degree  of  which  the  coefficients  are 
rational  symmetric  functions  of  the  elements  x^,  X2,  .  .  .  x^. 

From  the  equations  of  §  17, 

<p^  =  s,-\-  s,  V^,    <p,  =  s,  —  s,  \/l, 

we  have  for  the  elementary  symmetric  functions  of  <Pi  and  <p2 

<Pi  -h  <P2  =  2/Si, 

We  recognize  at  once  that  ^1  and  ^2  are  the  roots  of  the  equa- 
tion 

<p'  —  2s,<p-i-  {s;' .—  JS2')  =  0. 

It  is  however  to  be  observed  here  that  it  is  not  conversely  true 
that  every  quadratic  equation  with  symmetric  functions  of  the  x^Js 
as  coefficients  has  two- valued  functions,  in  the  present  sense,  as 
roots.  It  is  further  necessary  that  the  roots  should  be  rational  in 
the  elements  a?A.,  and  this  is  not  in  general  the  case. 


CHAPTER  11. 


MULTIPLE-VALUED  FUNCTIONS  AND  GROUPS  OF 
SUBSTITUTIONS. 

§  21.  The  preliminary  explanations  of  the  preceding  Chapter 
enable  us  to  indicate  now  the  course  of  our  further  investiga- 
tions, at  least  in  their  general  outline.  Exactly  as  we  have  treated 
one-valued  and  two-valued  functions  and  have  determined  those 
substitutions  which  leave  the  latter  class  of  functions  unchanged,  so 
we  shall  have  further,  either  to  establish  the  existence  of  functions 
having  any  prescribed  number  of  values,  or  to  demonstrate  their 
impossibility;  to  study  the  algebraic  form  of  these  functions;  to 
determine  the  complex  of  substitutions  which  leave  a  given  multiple- 
valued  function  unchanged;  and  to  ascertain  the  relations  of  the 
various  values  of  these  functions  to  one  another.  Further,  we  shall 
attempt  to  classify  the  multiple- valued  functions;  to  exhibit  them 
possibly,  like  the  two -valued  functions,  as  roots  of  equations  with 
symmetric  functions  of  the  elements  as  coefficients;  to  discover  the 
relations  between  functions  which  are  unchanged  by  the  same  sub- 
stitutions; and  so  on. 

§  22.  At  the  outset  it  is  necessary  to  devise  a  concise  notation 
for  the  expression  of  substitutions. 

Consider  a  rational  integral  function  of  the  n  independent  quan- 
tities a?i,  a?2,  ...  Xn,  which  we  will  denote  by  ^  (a?i ,  a^o?  •  •  •  x>).  If 
in  this  expression  we  interchange  the  position  of  the  elements  x^  in 
such  a  way  that  for  x^,  X2,  ...  x,,  we  put  X;^ ,  X;„,  .  .  .  o?,,^  respectively, 
where  the  system  of  numbers  i^ ,  i, ,  .  . .  ^„  denotes  any  arbitrary  per- 
mutation of  the  numbers  1,  2,  ...  n,  we  obtain  from  the  original 
function  <p  (x^,  Xo,  ...  x„)  the  new  expression  <p  (x,-^,  x,-,,  .  .  .  x,-^^). 

We  consider  now  the  manner  of  representing  by  symbols  such  a 
transition  from  a?i,  ccj,  ...  x^  to  Xi^,  X;^,  .  .  .  Xi^^ ;  to  this  transition 
we  have  already  given  the  name  of  substitution. 


CORRELATION  OF  FUNCTIONS  AND  GROUPS. 


19 


A.  In  the  first  place  we  may  represent  this  substitution  by  the 
symbol 

which  shall  indicate  that  every  element  of  the  upper  line  is  to  be 
replaced  by  the  element  of  the  lower  line  immediately  below  it.  In 
this  mode  of  writing  a  substitution  we  may  obviously,  without  loss, 
omit  all  those  elements  which  are  not  affected  by  the  substitution, 
that  is  all  those  for  which  x,,  =  Xi^ .  In  the  latter  case  the  entire 
number  of  elements  is  not  known  from  the  symbol,  but  must  be 
otherwise  given,  as  is  also  true  in  the  case  of  (f  itself,  since,  for 
example,  it  is  not  in  any  way  apparent  from  the  form  of 

<P  =  0-1  a?2  +  x^x^ 
whether  other  elements  Xr^,  x^^  ...  may  not  also  be  under  consider- 
ation, as  well  as  those  which  appear  in  (p. 

B.  Secondly,  we  may  make  use  of  the  result  of  the  preceding 
Chapter,  that  every  substitution  can  be  resolved  into  a  series  of 
transpositions.  If  we  denote  such  a  transposition,  i.  e.,  the  inter- 
change of  two  elements,  by  enclosing  both  in  a  parenthesis,  every 
substitution  may  be  written  as  a  series 

\X„Xj,)    {XcX(i)     .    .    .     {XpXq). 

This  reduction  can  be  accomplished  in  an  endless  variety  of 
ways.  For,  as  is  shown  in  §  14  of  the  preceding  Chapter,  we  can 
first  bring  any  arbitrary  element  to  its  proper  place,  and  then  pro- 
ceed with  the  remaining  n  —  1  elements  in  the  same  way.  Indeed 
we  may  introduce  any  arbitrary  transposition  into  the  series  and  can- 
cel its  effect  by  one  or  more  later  transpositions,  which  need  not 
immediately  follow  it  or  each  other. 

C.  Thirdly,  we  may  also  write  every  substitution  in  the  form 

\Xai  '^ao  '^cts    •   '   '    '^ay  \p^bi  '^bo  ^^63   •   •  •    '^b„J  V'^q  '^co  '^c^    •  •  •    "^c^/  •  •   •  •   • 

Here  each  parenthesis  indicates  that  every  element  contained 
in  it  except  the  last  is  to  be  replaced  by  the  next  succeeding,  the  last 
element  being  replaced  by  the  first.  The  parentheses  are  called 
cycles,  the  elements  contained  in  each  of  them  being  regarded  as 
forming  a  closed  system,  as  if  they  were,  for  example,  arranged  in 
order  of  succession  on  the  circumference  of  a  circle. 


20  THEORY    OF    SUBSTITUTIONS. 

If  we  wish  to  pass  from  the  notation  A  to  the  present  one,  the 
resulting  cycles  would  read 

yXi  Xi^  Xf.^    ■   •   •  )    ip^a^i^  '^i{       •   '   '    } 

Here  too  it  is  clear  that  those  elements  which  are  unaffected  by 
the  substitution,  every  one  of  which  therefore  forms  a  cycle  by 
itself,  may  be  omitted  in  the  symbol.  In  the  notation  A  these  ele- 
ments are  the  same  as  those  immediately  below  them. 

A  fourth  system  of  notation  which  is  indispensable  in  many 
important  special  cases  will  be  discussed  later. 

§  23.  It  is  obvious  that  each  of  the  three  notations.  A,  B,  and  C, 
contains  some  arbitrary  features.  In  A  the  order  of  arrangement 
of  the  elements  in  the  first  line  is  entirely  arbitrary;  in  B  the 
reduction  to  transpositions  is  possible  in  a  great  number  of  ways ;  in 
C  the  order  of  succession  of  the  cycles,  and  again  the  first  element 
of  each  cycle,  may  be  taken  arbitrarily. 

The  first  of  the  three  notations,  in  spite  of  its  apparent  sim- 
plicity lacks  in  clearness  of  presentation ;  the  second  is  defective,  in 
that  the  same  element  may  occur  any  number  of  times,  so  that  the 
important  question,  "  by  which  element  is  a  given  element  replaced," 
cannot  be  decided  at  first  glance,  and  the  equality  of  the  two  sub- 
stitutions is  not  immediately  clear  from  their  symbols.  We  shall,, 
therefore,  in  the  following  investigations,  employ  almost  exclusively 
the  representation  of  substitutions  by  cycles. 

The  following  example,  for  the  case  n  =  1,  will  serve  as  an  illus- 
tration of  the  different  notations. 

It  is  required  that  the  order  x^,  X2j  x^,  x^,  Xr^,  x^,  x^  shall  be 
replaced  by  the  order  x^,  Xt,  x^,  x^,  x^,  Xq,  X2, 
The  first  method  gives  us 

/  Ju-i    Jbo   *^3   *^4   *^5   *^ft   *^7    I     I    *^1    «^2   *^S   *^5-  ^7.   \ 

\  tX/Q  cA/'T  t^5  *^4  «^i   *^6  *^2-^  V.     3  vLj  vU^  JU-y   JL<^J  % 

By  the  second  we  have  variously 

\x-^x^  \Xy)c^  yx^x^)  =■  yXiX^)  [XiX^)  {X1X2)  {XiXj)  [X1X2) 

^^{X^X^j  {Xt^XQ)  (XlX^)  [X^X-,)  [XiXq)  (£C2'^7/  {f^iP^b)  \p^2'^i)  \^2'^&) 


Since  the  given  substitution  resolves  into  3,  5,  9,  .  .  . ,  transposi- 
tions, always  an  odd  number,  we  have  here  an  example  of  the  prin- 


COERELATION  OF  FUNCTIONS  AND  GROUPS.  21 

ciple  of   Chapter   I,  §  19,   aad   it  appears   that  this    substitution 
changes  the  sign  ol  \/  J. 
The  third  method  gives  us 

[XiX^X^)    \X2X'j)    \^i)    y^e)    —    [p^l'^'i^^)    \^2'^l)   —    \f^2'^l)    \'^3'^5'^l) 
=  {X'jX2)  {X^XiX^)  = 

§  24.  We  determine  now  the  number  of  all  the  possible  substi- 
tutions by  finding  the  entire  number  of  possible  permutations. 

Two  elements  x^,  x^^  can  form  two  different  permutations,  x^x^ 
and  ^20^1 .  If  a  third  element  x-^  be  added  to  these  two,  it  can  be 
placed,  1)  at  the  beginning  of  the  permutations  already  present 
X:iXiX2f  x.yjc^x^^  or  2)  in  the  middle:  x^x-^x^^  x.^x^x^^  or  3)  at  the 
end:  x^x^x^^^  XoX^x-^.  There  are  therefore  2  •  3  =  3!  permutations 
of  three  elements.  If  a  fourth  element  be  added,  it  can  occur  in 
the  first,  second,  third  or  fourth  place  of  the  3 !  permutations  already 
obtained,  so  that  from  every  one  of  these  proceed  4  new  permuta- 
tions. There  are  therefore  in  this  case  2  •  3  •  4  =  4!  permutations, 
and  again,  for  5  elements,  5 ! ,  in  general,  for  n  elements,  n !  permu- 
tations. 

If  now,  in  the  notation  A,  we  take  for  the  upper  line  the  natural 
order  of  the  elements,  x^,  x.,,  Xs,  ...  x„,  and  for  the  lower  line  suc- 
cessively all  the  n\  possible  permutations,  we  obtain  all  the  possible 
distinct  substitutions  of  the  n  elements. 

It  is  to  be  noticed  that  among  these  there  is  contained  that  sub- 
stition  for  which  the  upper  and  lower  lines  are  identical.  This 
substitution  does  not  affect  any  element;  it  is  denoted  by  1,  and 
regarded  as  unity  or  as  the  identical  substitution. 

Theorem  I.  For  n  elements  there  are  n\  possible  substitu- 
tions. 

To  obtain  the  same  result  from  the  notation  B  more  elaborate 
investigations  would  be  necessary  for  which  this  is  not  the  place;  in 
case  of  the  notation  C  it  is  easy  to  establish  the  number  n\  by  the 
aid  of  induction. 

We  arrive  in  the  latter  case  at  a  series  of  interesting  relations, 
of  which  at  least  one  may  be  noted  here.  If  a  substitution  in  the 
expression  for  which  all  the  elements  occur  contains 

a  cycles  of  a  elements,  b  cycles  of  /?  eleme^jts, 


22  THEORY    OF    SUBSTITUTIONS. 

where  aa-\-bi3-}-...=n,  N) 

we  can  obtain  from  this  by  rearrangement  of  the  order  of  the  cycles 
and  by  cyclical  permutation  of  the  elements  of  each  single  cycle 

a!  a"  6!/5'  .  .  . 

expressions  for  the  same  substitution.     Consequently  there  are 

n! 
a!  a"  6!  /5*  .  .  . 

distinct  substitutions  which  contain  a  cycles  of  a  elements,  b  cycles 
of  /?  elements,  and  so  on.  The  summation  of  these  numbers  with 
respect  to  all  possible  modes,  N),  of  distributing  the  number  n  gives 
us  all  the  possible  n\  substitutions.     Hence 

'^ 1 _-j   ^ 

§  25.  If  now  we  apply  all  the  n\  substitutions  to  the  function 
<p  (xi,  X.2,  ...  x„),  i.  e.,  if  we  perform  these  substitutions,  which 
may  be  denoted  by 

among  the  cci,  a?2,  ...  x,^  in  the  expression  (p,  we  obtain  n\  expres- 
sions, including  that  produced  by  the  substitution  Sj  =  1.  These 
expressions  we  may  denote  by 

or  simply,  where  no  confusion  is  likely  to  occur,  by 

S^l,    <P2,    <P3,     .   .   .    <Pa,    •   •   .    <Pn' 

These  values  are  not  necessarily  all  different  from  one  another. 
Some  of  them  may  coincide  with  the  original  value  <p  (x^,  X2,  ...  x„). 
We  direct  our  attention  at  first  to  the  complex  of  those  substitu- 
tions which  do  not  change  the  value  of  <p.  If  </>  is  symmetric 
this  complex  will  comprise  all  the  n\  substitutions;  if  ^  is  a  two- 
valued  function,  the  complex  will  contain  all  substitutions  which 
are  composed  of  an  even  number  of  transpositions,  and  only  these. 

Again,  for  example,  consider  the  case  of  four  elements  iCj,  0^2?  ^3>  ^4i 
and  suppose 

<p  yXi  ,^^2,^3,  X^)  ■=■  X   X2  -f'  X^X^. 
*Cauchy:  Exercices  ^'analyse,  III,  173. 


CORKELATION  OF  FUNCTIONS  AND  GROUPS.  23 

This  function  is  unchanged  by  8  of  the  possible  24  substitutions, 
namely  by 

^5  ^^    \XiXqX2X^)j        Sq  ^^  {XiX^X2X^),        §7   =   [XiX^)   {X2X^)J 
Oq  —  \JL-^JL'^)  !  t^2*^3/» 

By  the  remaining  4!  —  8  =  16  substitutions  v  is  changed,  and,  in 
fact,  is  converted  into  either 

We  note  then,  in  passing,  that  we  have  found  here  a  three-valued 
function  of  four  elements  which  is  unchanged  by  8  of  the  24  possi- 
ble substitutions  of  the  latter. 

§  26.  Those  substitutions  which  leave  a  function  (p(xi,X2,.  .  .  x^) 
unchanged^ the  number  of  which  we  shall  always  denote  by  r,  we 
shall  indicate  by 

6?)  s,  =  1,  S2,  Sg,  ...  s,; 

Si  =  1  is  of  course  contained  among  them.  Following  the  notation 
of  the  preceding  Section  we  have  then 

By  supposition  there  is  no  substitution  s'  different  from 
Si,So,S3,  ...  s,.,  which  leaves  the  value  of  <p  unchanged;  i.  e.,  we 
have  always 

^..ic.,     if  s'  +  s,      (A  =  1,2,  3,  ...r). 

If  now  we  apply  two  substitutions  Sa,  s^  of  our  series  Sj,  S2,  .  .  .  s^ 
successively  to  <p,  and  denote  the  result,  as  above  in  the  case  of  a 
single  substitution  by  <ps^  ^^ ,  then  since  <ps^  =  <p,  the  result  of  the 
two  operations  will  be 

and  from  this  we  conclude  that  SaS^  also  occurs  in  the  above  series 
6r).  Every  substitution  therefore  which  is  produced  by  the  succes- 
sive application  of  two  substitutions  of  G)  occurs  itself  in  G). 
What  is  true  of  two  substitutions  of  the  series  is  further  clearly 
true  for  any  number  whatever. 

The  substitution  which  results  from  the  successive  applicabion  of 
two  or  more  substitutions  we  call  their  product^  and  we  write  the 


24  THEOEY    OF     SUBSTITUTIONS. 

substitution  <y  which  produces  the  same  effect  on  the  order  of  the 
elements  a?i ,  a^o ,  ...  x,,  as  the  successive  application  of  Sa  and  s^s, 
as  the  product  o  —  s^  s^. 

The  product  of  any  number  of  the  operations  s  occurs  again  in 
the  series  G)  Sj ,  s,, ,  Sj. ,  ...  s,.  .  The  succession  of  the  operations  in 
a  product  (t  =  SaS^Sy  .  .  .  is  to  be  reckoned  from  left  to  right. 

§  27.     The  expression  of  such  a  product  in  the  cycle  notation 
which  we  have  adopted  is  obtained  as  follows : 
If  the  two  factors  of  a  product  are 

Sd   —  \p^a  '^ai  «^«2   •   •   •  /    \p^h  '^h'l  '^h^  •••)•••  j 
S/3  =^   {^b  ^bi  ^62   •    '   •   )    V^k  "^h  '^'k»   •■.)•••) 

then  in  §«  s^g  that  element  will  follow  x^  by  which  s^  replaces  x^^. 
Suppose,  for  instance,  that  this  element  is  x^,^ .  Again  in  Sa  s^  that 
element  will  follow  Xj,^  by  which  s^  replaces  Xj,, .  Let  this  be  for 
example,  x,,.^ ,  etc.     We  obtain 

SaSp  =  (X,,  Xj,^  Xj^..^  .  .  .  ) 

If  the  substitution  Sa  be  such  that  it  replaces  every  index  g  of 
the  elements  x^,  X2,  .  .  .  x^,  .  .  .  x,^,  hy  i,j,  and  if  s^  be  such  that  it 
replaces  every  index  ghj  k^,  or,  in  formulae,  if 

Sa  ^^  [Xi  CCj-j  X/^^  '  '  •  )   \Xa  Xi„  ...   j  ...  , 

Sp={XiXj,^X;,^^  . . . )  {x,Xk^ ...)..., 
then  the  product  will  be  of  the  form 

Sa  s^  =  (xi  Xj,.,^  .  .  .  )  (a?,  Xj,;^  ...)... 
The  following  may  serve  as  an  example: 

Sa  =  {X^X^X^  {x^Xr) ,       S^  =  (x^X^Xq)  (x^Xr^, 

Sa  s^  =  (x^x^)  {x^x^x^x^  (a^g)  =  {x^x^  {x.^x^x^x^ . 
We  have  introduced  here  the  expression  "product."     The  ques- 
tion now  arises  how  far  the  fundamental  rules  of  algebraic  multipli- 
cation 

a  ■  b  =  b  •  a,     a  •  (b  •  c)  =  {a  -  b)  ■  c 

remain  valid  in  this  case.  An  examination  of  this  matter  will  show 
that  the  former,  the  commutative  law,  in  general  fails,  while  the 
second,  the  associative  law,  is  retained.    In  fact  the  multiplication  of 


CORRELATION  OF  FUNCTIONS  AND  GROUPS.     '         25 

as  performed  above  shows  that  it  is  only  in  the  special  case  where, 
for  every  a  ,  4^  =  A;,^,  that  the  order  of  the  two  factors  ««  and  s^  is 
indifPerent.  This  occurs,  for  example,  as  is  a  priori  clear,  if  the 
expressions  for  Sa  and  s^  contain  no  common  elements. 

We  may  therefore  interchange  the  individual  cycles  of  a  substi- 
tution in  any  way,  since  these  contain  no  common  elements.  In 
the  notation  B  of  page  19,  on  the  other  hand,  this  is  not  allowable. 

Passing  to  the  associative  law,  however,  if 

Sa.  =  {x,  Xi,  ...)..  .,    S^  —  {x,Xk, )  .  .  . ,    Sy—  {XJ)C,^  ...)..., 

we  have  the  following  series  of  products, 

S/3  Sy  ^  [Xg  Xjj^^    ...)...,     '      SaS^^=  Xi,  Xj..^  ...)... 
Sa{s^Sy)   ={XgX^^.^   ...)..  .,  (SaS^)Sy=   (x^Xj,.^   ...)..., 

from  which  follows 

Theorem  II.  In  the  multiplication  of  substitutions  a  col- 
lection  of  the  factors  into  sub-products  without  change  in  the  order 
of  the  factors,  is  permissible.  An  interchange  of  the  factors,  on 
the  contrary,  generally  alters  the  result.  Such  an  interchange  is 
however  permissible  if  the  factors  contain  no  common  elements. 

§  28.  From  the  preceding  developments  it  appears  that  those 
substitutions 

G)        ^  Sl  =  1,S2,S3J  .  .  .  Sa,  .  .  .  s, 

which  leave  a  given  function  <p  {x^^Xo,  ...  x,)  unchanged,  form  a 
closed  group  in  this  respect,  that  the  multiplicative  combination  of 
its  substitutions  with  one  another  leads  only  to  operations  already 
contained  in  the  group. 

The  name  ^^group^^  *  we  shall  always  use  to  denote  a  system  of 
substitutions  which  possesses  this  characteristic  property  of  repro- 
ducing itself  by  multiplication  of  its  individual  members.  The 
number  of  elements  operated  on  is  called  the  degree  of  the  group. 
It  is  not  however  necessary  that  all  the  elements  should  actually 
occur  in  the  cycles  of  the  substitutions.     Thus 

♦Cauchy,  who  gave  the  first  systematic  presentation  of  theTlieory  of  Substitutions 
in  the  Exercices  d'Analyse  et  de  Physique  Math^matique,  employs  the  name  "  system  of 
conjugate  substitutions:'  Serret  retains  this  name  in  his  Algebra.  The  shorter  name, 
*'  group,"  was  introduced  by  Galois. 


26  THEOEY    OF    SUBSTITUTIONS. 

Si  ^^^  J.  J  §2  —  \OC13C2J  {OC^OC^^ 

form  a  group,  for  we  have  Sj  S2  =  SoS^  =  S2,  S2  So=  s^  =  1.  This 
group  is  of  degree  4,  if  only  the  elements  x^ ,  X2 ,  x.^,  x^  be  under 
consideration.  But  we  might  also  regard  the  group  as  affecting,  for 
example,  6  elements  a^^ ,  a?2  j  •  •  •  ^e  j  iii  which  case  we  may,  if  desired, 
write  for  §2 

S.;  =  {X,X2)  (XoXi)  (x,)  (x^). 

The  degree  of  the  group  is  then  6. 

The  number  of  substitutions  contained  in  a  group  is  called  its 
order;  as  already  stated,  this  number  will  always  be  denoted  by  r. 

The  entire  system  of  substitutions  which  leave  the  value  of  a 
function  <p  (Xi,  X2,  ...  Xn)  unchanged  is  called  the  group  of  substi- 
tutions belonging  to  the  function  c^,  or,  more  briefly,  the  group  of 
(p.  The  degree  of  the  group  expresses  the  number  of  elements 
Xi ,  a-o ,  .  .  .  X,,  under  consideration ;  its  order  gives  the  number  of 
substitutions  which  leaves  the  function  ^  unchanged. 

Thus,  given  the  four  elements  x^ ,  X2 ,  x^ ,  x^ ,  and  the  function 
<p  =  a?i  0^2  -{-  x^x^, 
the  degree  of  the  group  belonging  to  ^  is  4;  its  order,  as  shown  in 
§  25,  is  8. 

For  the  five  elements  x^,  a?2,  x^,  x^,  x^,  the  same  function  <p  has 
a  group  of  degree  5  and  of  order  8,  identical  with  the  preceding 
one. 

For  the  six  elements  x^,  X2,  x-^,  .  .  .  x^,  the  same  ^has  a  group  of 
degree  6.  To  the  group  above  we  must  now  add  all  those  substitu- 
tions which  arise  from  combining  the  former  with  the  interchange  of 
3^5  and  Xf^.  The  group  now  contains  beside  the  eight  substitutions  of 
§  25  the  following  eight  new  ones: 

Sg  =  (X^X^),      Sio  =  (X1X2)  {Xr,X^),      S„  =  (0:^3^4)  (^s^e), 

S12  =  {X,X2)  (X^X^)  {x^Xq\      Si3  =  {XiX.sX2Xi)  (x^Xq),     Su=  {XiX^X2X.^)  (Xr.X^), 

Si5  =  (XiX^)  {X2Xi)  (X^X^\       S16  =  (XiXi)  {X2X3)  (Xr^X^)  . 

The  order  of  the  group  of  <p  is  therefore  now  8  •  2  =  16. 

It  is  easy  to  see  that  if  we  regard  <p  as  dependent  on  n  >  4  ele- 
ments, the  order  of  the  corresponding  group  becomes  8  •  (n  —  4) ! , 
the  group  being  obtained  by  multiplying  the  8  substitutions  of  §  25 
by  all  the  substitutions  of  the  elements  Xr^,  x^,  .  .  .  x„: 


CORRELATION  OF  FUNCTIONS  AND  GROUPS.  27 

§  29.     The  following  theorem  is  obviously  true : 

Theorem  III,  For  every  single-  or  multiple -valued  func- 
tion there  is  a  group  of  substitutions  which,  applied  to  the  function^ 
leave  it  unchanged. 

To  show  the  perfect  correlation  of  the  theory  of  multiple-valued 
functions  and  that  of  groups  of  substitutions  we  will  demonstrate 
the  converse  theorem : 

Theorem  IV.  For  every  group  of  substitutions  there  are 
functions  ivhich  are  unchanged  by  all  the  substitutions  of  the  group 
and  by  no  others. 

We  begin  by  constructing  a  function  ^  of  the  7i  independent 
elements  x^,  X2,  .  .  .  x,^  which  shall  take  the  greatest  possible  num- 
ber of  values,  viz :  n ! ;  ^  is  therefore  to  be  changed  in  value  by  the 
application  of  every  substitution  difPerent  from  unity. 

Taking  n  -\-l  arbitrary  and  difPerent  constants  «o,  a^,  .  .  .  a.,,  we 
form  the  linear  expression 

^  =  «0  +  «i  0^1   +  «2  +    •  •   •    +   ««  ^« 

If  now  two  substitutions  Sa=(x^,0Ci^ ...)...  and  s^={XsX^,^ ...)..., 
on  being  applied  to  <p,  gave  the  same  result,  we  should  have 

0  =  <Psa  —  <Psp  =  «i(i^H  —  JO,.,)  +  a^ix,,  —  X^„)-\- 

But  the  a^'s  being  arbitrary  quantities,  this  equation  can  only  be 
satisfied  if  each  parenthesis  vanishes  separately  (§  2,  C),  that  is  we 
must  have  Xi^=Xj,y  But,  if  this  be  the  case,  since  the  x^^a  are  also 
independent  quantities,  the  two  substitutions  Sa  and  s^  both  replace 
every  X\  by  the  same  element  x,-^  =  a?^^,  so  that  Sa  and  s^  are  identi- 
cally the  same.  It  is  only  in  this  case,  where  Sa  and  s^  are  identical, 
that  they  can  produce  from  ^  the  same  value.  Accordingly  <p  has  n  [ 
distinct  values. 

§  30.  If  now  a  group  G  be  given,  composed  of  the  substitu- 
tions 

Si  ^    1,    S^,    S-^,     .  .  .     Sa,      .  .  .     Sy, 

which  we  will  indicate  symbolically  by  the  equation 

Cr  :=    ySi  ,    §2 )    ^3    •   •   •     ^a  5   •   •   •    S/J  5 

we  apply  all  the  substitutions  of  G  to  the  72!-valued  function  with 
n  -\-  1  parameters 


28  THEORY    OF    SUBSTITUTIONS. 

and  denote  the  results  of  these  operations  as  before  by  correspond- 
ing subscripts  attached  to  the  f^'s 

Then  the  product  of  these  functions  <p 

^  =  ^\  •  <P.,  •  <Ps,  .  .  .  <fs,. 
will  be  one  of  the  functions  to  which  the  group  of  substitutions  G 
belongs. 

To  prove  this  it  must  be  shown  1)  that  .f  is  unchanged  by  every 
substitution  (t  of  G,  and  2)  that  ^  is  changed  in  value  by  every 
substitution  r  which  does  not  occur  in  G. 

In  regard  to  the  first  condition  we  have 

0       (D  •    <P  •    (D  Cf'         ' 

^ (T         Tsio-       r s.2<T       Vsyo-  •  •  •    V-s,cr5 

and,  from  the  definition  of  a  group,  s-^(t  =  <r,  Sr,<7,  S;,(t,  .  .  .  s,.<7  are  again 
contained  in  G.  Moreover,  these  products  are  all  different  from  one 
another;  for  if  Sa'^  and  s^t  applied  to  <p  have  the  same  effect,  this 
must  also  be  the  case  with  Sa  and  s^  alone,  and  therefore  Sa  and  s^ 
are  identical.  Accordingly  the  substitutions  s^fr—ff,  s.^f^,  SgT,  .  .  .  s,.<t 
are  identical,  apart  from  their  order,  with  Sj  =  1,  s.,,  Sg,  ...  s,.,  and 
hence  the  functions 

<Psia^   <P,.<Tj   9s^iTi    '  •  '    <fs,.<r 

are  identical  with 
and  accordingly 


9s,,  <Ps,,  <P.,,    . 


as  was  asserted. 

As  to  the  second  condition,  the  substitutions  s^r ,  SoT,  s^t^  .  .  .  s,.t 
are  all  different  from  Sj ,  83 ,  Sg ,  .  .  .  s,.,  and  consequently  the  func- 
tions 9^„jT,  ^*,Tj  S^-syTj  .  •  •  9'vT  ^^'®  ^11  different  from  the  factors 
f%i)  9..0,  ^«3,  .  .  .  ^«,.  of  (f^.  Moreover,  this  difference  is  such  that  no 
<PsiT  can  be  equal  to  the  product  of  a  <p,,^.  by  a  constant  c, ,  in  which 
case 

would  become  equal  to  ^  if  c,  a,  Cg  .  .  .  c,.  =  1.     For,  if 

«o  +  «i  OCr,  +  ttaO?^,  +  .  .  .  =  C,-(ao  +  «i  a-/,  +  aoa\-,  +  ...), 


CORRELATION  OF  FUNCTIONS  AND  GROUPS.  29 

then,  since  the  «a's  are  arbitrary  quantities,  it  would  follow  at  once 

that 

c,  =  1, 

and  consequently  we  should  have  the  impossible  equation  (fr  =  (psf. 

§  31.  In  many  cases  the  calculation  of  (^  is  impracticable,  since 
the  multiplication  soon  becomes  unmanageable  even  for  moderately 
large  values  of  r.  There  is  however,  another  process  of  construc- 
tion in  which  the  product  is  replaced  by  a  sum,  and  every  difficulty 
of  calculation  is  removed. 

We  begin  by  taking  as  the  basis  for  further  construction,  instead 
of  the  linear  function  (f,  the  following  function 

^  (a?i ,  a^-o ,  a^3 ,  ...  X,)  =  Xi'^i  Xo'^^  x-i""^  .  .  .  x,>. 

The  «^'s  are  to  be  regarded  here,  as  before,  as  arbitrary  quanti- 
ties, and,  as  the  ^a's  are  also  arbitrary,  it  follows  at  once  that  9''  is 
an  w!-valued  function.     For,  if 

then  we  must  have  identically 

x^^^x/'^x/^ . . .  x^^n  =  x{y^ x-^y^ x^y^ . . .  x,yn, 

and,  from  §  2,  C,  this  is  only  possible  if  every  ,3;  is  equal  to  the  cor- 
responding )^f^  that  is,  if  the  substitutions  (t  and  r  are  identical. 

We  denote  the  functions  which  proceed  from  v''  under  the  opera 
tion  of  the  substitutions  of  G  by 

d>    —  d'     d'       (!>  d> 

and  form  now  the  sum 

^■=^'s,    +</'«,  +  </'s3+    ...     +0S,. 

The  proof  of  the  correlation  of  G  and  4'  proceeds  thien  exactly  as 
in  the  preceding  Section  in  the  case  of  G  and  ^. 

Remark. — By  making  certain  assumptions  with  respect  to  the 
a's  we  can  assign  to  the  0's  some  new  properties.  Thus  we  may 
select  the  a's  in  such  a  way  that  an  equation  between  any  two  arbi- 
trary systems  of  the  a's, 

«n  +  «^  + .  •  .  «/x  =  "^-1  +  "-^-2  +  •  •  .  «;fc^ , 
necessarily  involves  the  equality  of  the  separate  terms  on  the  right 
and  left.     This  condition  is  satisfied  if,  for  example,  we  take 


30 


THEORY    OF    SUBSTITUTIONS. 


«2   >    «1,        «3    >    «1  +   «2,         «4   >    «1   +  «2  +   «3  , , 

in  particular  if 

a^—1,      as  =  2,      a3  =  4,      a,  z=z  8,      «5  =  16, 

E.  g.    If  a,j  +  a._^  -|-  _  .  -^  a.^  =  13^  ^e   must   have  i^  =  1,  u  =  3, 
4  =  4,     /  =  3. 

Example. — We  will  apply  the  two  methods  given  above  to  the 
familiar  group 

G  =  [l,    (x.QCo),    (x,x,),    (x.x,)  (_x,x,),    {x,x,X^23c,),    (x,Xi)  (x,x,), 
(XjX.,X2Xi),  {x^x^x^Xo)'],  {n  =  4:,  r  =  8) 

taking  as  fundamental  functions 

<p  =  Xi-\-  ix.  —  Xs  —  ix^  and   (J'  =  Xi^X2^xJ^Xi^,    . 


where,  as  usual,  i  =  V — 1. 

We  have  then  the  following  results: 

^  =    («^i  +  i^2  —  ^3 — ■^'^4)  (^2  +  i^i  —  ^3 — i^d  (^1  +  «^2 — ^i  —  ^"^3) 
(xs  +  ^a■4 — a:2 — *'^i)  (^4  +  *'a:'3 — ^i  —  ^'^2) 

=    [(^1  +  ^'^2 ^3 ^^i)   (^2  H~  ^''^l ^3 ^'«^4)   (^1  +  i^2 ^4 i^s) 

(a?2  +  *a?i — Xi — ^^3)]^ 

■ilf  /y>   /y>  •^^y»  °      I       /y»    'y«  •^,-yi   "      I       /y»   ^y»  "^y,  o      I       /y»   ^y«  2^y,  3      I       /y.   /y.   2-,  O      |       ^y,   /y,  2^y,  3 

^       eAy2«^3    «^^  I       eA^JtA^g    «A^^  j       «A-2«^'4    «^3  |       eA^^jM^'^    M>g  |       eA^^eX/j    «A^2  r       eA-qeA^2    *^1 

~p"  «//^U?2  «^i    ~)~  t3-'3U?j  y>2 
=  (^1  +  ^2)  (^3  V  +  ^4^')  +  fe  +  3C,)  {Xlx./  +  ^-2^') 
^^^^   {^Xi  "T~  ^2}  \'^3  "!     ^i)  \p^9  '^i    "T~  •^l  ^^2  /• 

Neither  of  the  two  methods  furnishes  simple  results  directly. 
But  from  ^  >ve  may  pass  at  once  to  the  function 

[{Xi  —  x^y  +  (a?3  —  a?,)-]  [{x,  —  x,f  +  (x,  —  x,y] , 

and  from  ^F  to  the  two  functions 

(a?i  +  X2)  (x^  -\-  Xi)     and     XiXo-j-  x^x^ , 

the  latter  being  already  known  to  us.  It  is  clear  also  that  by  alter- 
ing the  exponents  which  occur  in  '/^  we  can  obtain  a  series  of  func- 
tions all  of  which  belong  to  G.  Among  these  are  included  all 
functions  of  the  form 

(/y»   O       I        ry*   o\    ( r^  O.       I       /y»   a\  /y«   O   /yi   O       I        /y»   tt  ^y»   tt 

X\        1^  •^2   /   \'^3        I     '*'4   /  ?        Xj     ^2     n^  t^3     t/.4   . 


CORRELATION    OF    FUNCTIONS    AND    GROUPS.  31 

In  general  we  perceive  that  to  every  group  of  substitutions  there 
belong  an  infinite  number  of  functions. 

It  may  be  observed  however  that  we  cannot  obtain  all  functions 
belonging  to  a  given  group  by  the  present  methods.  Thus  the 
function 

belongs  to  the  group 

G  =  [1,  (xix,),  (x^x,),  (xiX,)  (xo,  a?,)], 

but  cannot  be  obtained  by  these  methods.  More  generally,  if 
the  functions  v^'',  V'",  v''"'?  •  •  •  belong  respectively  to  the  groups 
H',  H",  H'",  .  .  .  ,  and  if  the  substitutions  common  to  these  groups 
(Cf.  §  44,  Theorem  VII)  form  the  given  group  G,  then  the  function 

where  the  a's  are  arbitrary,  belongs  to  the  group  G. 

§  32.     We  now  proceed  to  consider  the  case  where  the  elements 
x^,X2,  .  .  .  x„  are  no  longer  independent  quantities. 

Theorem  V.  Even  where  any  system  of  relations  exists 
among  the  elements  Xi,X2,  ...  x„ ,  excluding  only  the  case  of  the 
equality  of  two  or  more  elements,  we  can  still  construct  n\-valued 
functions  of  x^,  x^,  .  .  .  Xn.^ 

Using  the  notation  of  the  preceding  Section,  we  start  from  the 
same  linear  function 

cp    —a^-\-  a^x^  -f  a^X.  +  .  .  .  +  a„a?„ 

and  form  the  product  of  the  differences  of  the  ^'s 

n\{n\ 1) 

this  product  being  taken  over  the       \     c, —    possible   combinations 

of  the  ^'s  in  pairs.     Expanding  we  have 

In  no  one  of  these  factors  can  all  the  parentheses  vanish,  since 
otherwise  either  the  substitutions  fff  and  r,:  must  be  identical,  or  else 
the  x'q  are  not  all  different.  The  product,  regarded  as  a  function  of 
the  a's ,  therefore  cannot  vanish  identically  (§  2,  D).    Consequently, 

*C/.  G.  Cantor:  Math.  Annalen  V,  133;  Acta  Math.  I.  372-3. 


32  THEORY    OF    SUBSTITUTIONS. 

(§  2),  there  are  an  infinite  number  of  systems  of  values  of  the  a's 
for  which  all  the  n\  values  of  <p  are  different  from  one  another. 

Corollary.  The  oyily  relations  a7nong  the  x^s  ivhieh  can 
determine  the  equality  of  linear  expressions  of  the  form 

c?  =  «o  -f-  a^Xi  -j-  a^X.2  +  .  .  .  +  o-nXn 

independently  of  the  values  of  the  a's,  are  the  relations  of  equality 
of  two  or  more  x^s. 

§  33.  With  the  Theorems  III  and  IV  the  foundation  is  laid  for 
a  classification  of  the  integral  functions  of  n  variables.  Every  func- 
tion belongs  to  a  group  of  substitutions;  to  every  group  of  substi- 
tutions correspond  an  infinite  number  of  functions.  This  relation- 
ship is  not  the  only  connection  between  functions  which  are 
unchanged  by  the  same  substitutions;  we  shall  find  also  a  corres- 
ponding algebraic  relation,  namely,  that  every  function  which 
belongs  to  a  group  G  can  be  rationally  expressed  in  terms  of  every 
other  function  belonging  to  the  same  group. 

It  becomes  then  a  fundamental  problem  of  algebra  to  determine 
all  the  possible  groups  of  substitutions  of  n  elements.  The  general 
solution  of  this  problem,  however,  presents  difficulties  as  yet  insup- 
erable. The  existence  of  functions  which  possess  a  prescribed 
number  of  values  is  discussed  in  one  of  the  following  Chapters.  It 
will  appear  that  there  are  narrow  limits  to  the  number  of  possible 
groups.  For  example,  in  the  case  of  7  elements,  there  is  no  function 
which  possesses  3,  4,  5,  or  6  values ;  and  we  shall  deduce  the  general 
proposition  that  a  function  of  n  elements  which  has  more  than  twa 
values,  will  have  at  least  n  values,  if  n  >  4.  A  series  of  other  anal- 
ogous results  will  also  be  obtained. 

For  the  present  we  shall  concern  ourselves  only  with  the  con- 
struction and  the  properties  of  some  of  the  simplest,  and  for  our 
purpose,  most  important  groups.  * 

§  34.  First  of  all  we  have  the  group  of  order  7i ! ,  composed  of 
all  the  substitutions.  This  group  belongs  to  the  symmetric  func- 
tions, and  is  called  the  symmetric  group. 

In  Chapter  I  we  have  seen  that  every  substitution  is  reducible 
to  a  series  of  transpositions.     Accordingly,  if  a  group  contains  all 

•  Cf.  Serret:  Cours  d'alg^bre  sup6rieure.  II,  §§  416-429.    Cauchy :  loc.  cit. 


COREELATION  OF  FUNCTIONS  AND  GROUPS.  33 

the  transpositions,  it  contains  all  the  possible  substitutions  and  is 
identical  with  the  symmetric  group.  To  secure  this  result  it  is  how- 
ever sufficient  that  the  group  should  contain  all  those  transpositions 
which  affect  any  one  element,  for  example  x^ ,  that  is  the  transposi- 
tions 

.{OCiXo),     [XiX^),    {XlX^),  .  .  .  [XiX,i). 

For  every  other  transposition  can  be  expressed  as  a  combination 
of  these  n^l-,  in  fact  every  (XaXp)  is  equivalent  to  a  series  of  three 
of  the  system  above, 

(XaXp)   =    (x^Xa)  (XiXp)  (x^Xa), 

(where  it  is  again  to  be  noted  that  the  order  of  the  factors  is  not 
indifferent).     We  have  then 

Theorem  VI,     A  group  of  n  elements  Xi,  x^,  .  .  .  x„  which 
contains  the  n  —  1  transpositions 

is  identical  with  the  symmetric  group. 

Corollary.     A  group  ivhich  contains  the  transpositions 

(XaXp),    (XaXy),    .  .  .   {Xa,X^) 

contains  all  the  substitutions  of  the  symmetric  group  of  the  elements 

Xd  ,      «A/|3  J      Xy  ,        .      .      .       X^. 

§  35.  We  know  further  a  group  composed  of  all  those  substitu- 
tions which  are  equivalent  to  an  even  number  of  transpositions.  For 
all  these  substitutions,  and  only  these,  leave  every  two-valued  func- 
tion unchanged,  and  they  therefore  form  a  group.  We  will  call 
this  group  the  alternating  group.  Its  order  r  is  as  yet  unknown, 
and  we  proceed  to  determine  it.     Let 

L)  Sj  =  1,  §2  ,  S3,    ...  s,. 

be  all  the  substitutions  of  the  alternating  group,  and  let 

ii)  Si  ,    S.f  ,    S^  ,     .  .  .     Sf 

be  all  the  substitutions  which  are  not  contained  in  I),  and  which  are 
therefore  composed  of  an  odd  number  of  transpositions.  We  select 
now  any  transposition  <t,  for  example  o-  =  {X1X2),  and  form  the  two 
series 

r)  Si<T,    SoT,    SgfT,      ...     S,.(T, 

IT)  s/(Tj    S^'ff,    S/(T,     ...     s/iT. 


34  THEOEY    OF    SUBSTITUTIONS. 

Then  every  substitution  of  I')  is  composed  of  an  odd  number  of 
transpositions,  and  every  substitution  of  II')  of  an  even  number. 
Consequently  every  substitution  of  I')  is  contained  in  II),  and  every 
one  of  II')  is  contained  in  I) .  Moreover,  Sa^  ^  s^tr,  and  Sa'(T  ^Sprr, 
for  otherwise  we  should  have 

Sa  =  Sa  (<y  •   <^)  =   (Sa'y)  ^  =   (S|3  ^)  '^  =:   Sp{(7  •   (t)   =  Sp, 
Sa    =  Sa'{T  .  (t)  =   {Sa'(T)  ^  :=   (.S^V)  G  =  Sp' {<T  •   <t)  =z  S^\ 

since  (t  .  a  —  (^Ti^Ca)  (^1^2)  =  1- 

It  follows  from  this  that  r ^t  and  r^t,  that  is  r  =  ^.  Again, 
since  I)  and  II)  contain  all  the  substitutions,  r-\-  t  =  7i\.     Hence 

n\ 

We  will  note  here  that  there  is  no  other  group  F  of  order  ^  • 
For  a  function  ^1  belonging  to  such  a  group  would  be  unchanged  by 
^  substitutions,  and  would  be  changed  by  all  others.  It  would 
therefore  possess  other  values  beside  <pi .  Suppose  ^o  to  be  one  of 
these  values,  and  let  o-  be  a  substitution  which  converts  <Pi  into 
<P2  =  <PcT.     If  now  <px  is  unchanged  by  the  group 

HI)  r=[l,s/,  83',  ...8'i»!], 

then  <pi  must  be  converted  into  <P2  by  all  the  substitutions 

IV)  <T,  s/tT,  S;/(r,  .  .  .  s'i„.o-; 

for  Sa'  leaves  <Pi  unchanged  and  <t  converts  <Pi  into  (p., ,  consequently 
S\(T  will  also  convert  <Pi  into  9^2-  Again  all  the  substitutions  Sa<7  of 
the  series  IV)  are  different  from  one  another.  For,  if  Sa'(T  =  Sp'<T, 
it  would  follow  that  Sa'  =  s^'.  The  substitutions  s^'ff  are  also 
different  from  the  SA"s,for  the  latter  have  a  different  effect  on  <fi 
from  the  former.  Consequently  III)  and  IV)  exhaust  all  the  possi- 
ble substitutions,  and  ^1  is  therefore  a  two -valued  function,  for  there 
is  no  substitution  remaining  which  could  convert  <pi  into  a  third 
value.     The  group  F  is  therefore  the  alternating  group. 

Theorem  VII.  For  n  elements  there  is  only  one  group  of 
order  -j .  This  is  the  alternating  group.  It  belongs  to  the  two- 
valued  functions. 

We  can  generalize  this  proposition.  The  proof,  being  exactly 
parallel  to  the  preceding,  may  be  omitted. 


COERELATION  OF  FUNCTIONS  AND  GROUPS.  35 

Theorem  VIII.  Either  all,  or  exactly  half  of  the  substi- 
tutions of  every  group  belong  to  the  alternating  group. 

Corollary.  Those  substitutions  of  any  given  group  of  order 
r  which  belong  to  the  alternating  group,  form  a  group  within  the 
given  group,  the  order  of  which  is  either  r  or  -^-. 

The  simplest  substitutions  belonging  to  the  alternating  group 
contain  three  elements  in  a  single  cycle,  {XaX^Xy).  They  are  equiv- 
alent to  two  transpositions,  (XoX^Xy)  =  {XaX^  (xpXy), 

A  substitution  containing  only  one  cycle  (a?,^,  o?^,  ...  x^J  we 
shall  call  a  circular  substitution  of  order  m. 

Theorem  IX.  If  a  group  of  n  elements  contains  the  n  —  2 
circular  substitutions 

{XiX2X^),  {XiX2X^),    .  .  .    {XiX2X„)j 

it  is  either  the  alternating  or  the  symmetric  group. 
For  since 

yXgiXsXy)  —  \X^X2Xq)   ( iA/jt«/2«^/3/   \X^X2Xyf   [X^XoXfi)   ( «X;jtt/2«^a/   \,**^i'^.)"-'/3/) 

it  follows  that  the  given  group  contains  every  circular  substitution 
of  the  third  order.     And  again,  since 

it  follows  that  all  substitutions  occur  in  the  given  group  which  are 
composed  of  two,  and  consequently  of  four,  six,  or  any  even  num- 
ber of  transpositions.     The  theorem  is  therefore  proved. 

We  add  the  following  theorem: 

Theorem  X.  If  a  group  contains  all  circular  substitutions 
of  order  m  +  2,  it  will  contain  also  all  those  of  order  m,  and  con- 
sequently it  will  contain  either  the  alternating  or  the  symmetric 
group,  according  as  m  is  odd  or  even. 

For  we  have 

\J(j-)^Ju2    ...  ^fi}*^(t*^h)    KX-tJLt}    ...    tXj^^X^^  Xf)  I    yX-ff^Xjf^ , 1    ...    X2XffX^Xff\ 

—  (^^lCC2   .  .  .   X,„  _  1  X,„j. 

Finally,  we  can  now  give  the  criterion  for  determining  whether 
a  given  substitution,  expressed  in  cycles,  belongs  to  the  alternating 
group,  or  not.     The  proof  is  hardly  necessary. 


36 


THEOKY    OF    SUBSTITUTIONS. 


TlieoreiXl  XI.  If  a  substitution  contains  m  elernents  in  k 
cycles^  it  does  or  does  not  belong  to  the  alternating  group  according 
as  m  —  A;  is  even  or  odd. 

§  36.  Any  single  substitution  at  once  gives  rise  to  a  group,  if  we 
multiply  it  by  itself  i.  e.,  if  we  form  its  successive  powers.  Tbe 
meaning  of  the  term  "power"  is  already  fully  defined  by  the  devel- 
opments of  §  27.     We  must  have 

g,.   _  g,«-l    .  g  _  ^   .  g,«-l  _  ^,«-2   .  ^2  _  g2    .  g,„_2  ^  ga   .  g.«-a 
=  §«   .  S^    .  g'"-a-^  _ 

The  process  of  calculation  of  the  powers  of  a  substitution  is  also 
clear  from  the  preceding  Sections.  If  we  wish  to  form  the  second, 
third,  fourth,  .  .  .  «*'»  power  of  a  cycle,  or  of  any  substitution,  we 
write  after  each  element  the  second,  third,  fourth,  .  .  .  «"^  following 
element  of  the  corresponding  cycle,  the  first  element  of  each 
cycle  being  regarded  as  following  the  last.  Thus,  from  the  cycle 
{x^x^x^x^Xr^  .  .  .  )  we  obtain  for  the  second  power  (x^x^Xr^  •  •  •  )?  ^^^  ^-^^ 
third  (x^x^x-,  .  .  .  ),  for  the  fourth  (x^Xr^x.:,  ■  -  •  ),  ©tc.  It  is  obvious 
that  in  this  process  a  cycle  may  break  up  into  several.  This  will 
occur  when  and  only  when  the  number  of  elements  of  the  cycle  and 
the  exponent  of  the  power  have  a  common  divisor  d  >  1.  The 
number  of  resulting  cycles  is  then  equal  to  d. 

For  example, 

[XiX2X^X^X^X^)     =:    {XiX^Xt^j   {X2X^X(^)j 
yXlX2X^X^Xr,XQ)     ^=   [XiX^j  [XoX^j  yX^X^j^ 
{XiX2X^X^Xr^XfJ     ^    \XiXr,X-^)  {X2X(^X^), 

(/y   r^   /y   r^   /y   r^   \''     / /y   /y    .'yt   ry   r^   q^   \ 

If  the  number  of  elements  of  a  cycle  be  m,  then  the  m^^\  (2w)*^, 
(3m)^'\  .  .  .  powers  of  the  cycle,  and  no  others,  will  be  equal  to  1. 

E.  g.       {x^X2X'^X^Xr,X^'^   —   {XyX2X-;.X.^Xr,X^^'^  =    ...    =1. 

If  a  substitution  contains  several  cycles  -with  m^ ,  m, ,  mg , .  .  . 
elements  respectively,  the  lowest  power  of  the  substitution  which 
is  equal  to  1  is  that  of  which  the  exponent  r  is  the  least  common 
multiple  of  m, ,  mj ,  w^ ,  .  .  .       Thus 

[{x,X2X.i)  {x.Xr)  {XfflCT)J  =  1. 

This  same  exponent  r  is  also  the  order  of  the  group  formed  by 
the  powers  of  the  given  substitution.     For  if  we  calculate 


CORRELATION  OF  FUNCTIONS  AND  GROUPS.  37 

SfSj,,,S         jO  —  J.  J 

a  further  continuation  of  the  series  gives  merely  a  repetition  of  the 
same  terms  in  the  same  order: 

s'-+'  =  s,  s^-+'  =  s\  s^-+'  =  sV  .  .  s''-'  =  s>-\  s'^-  =  s'=l,  ... 

Moreover  the  powers  of  s  from  s^  to  s'  are  different  from  one 
another,  for  if 

then  we  should  have  contrary  to  hypothesis 
s'*  =  1     (//  <  r). 

The  extension  of  the  definition  of  a  power  to  include  the  case  of 
negative  exponents  is  now  easily  accomplished.     We  write 

SO  that  we  have 

gfc  g-fc  _  2. 

The  substitution  s^  therefore  cancels  the  effect  of  the  substitu- 
tion s~%  and  vice  versa.  The  negative  powers  of  a  substitution  are 
formed  in  the  same  way  as  the  positive  powers,  only  that  in  forming 
( — 1)^^  ( — 2)^,( — 3)^,  .  .  .  powers,  we  pass  backward  in  each  cycle 
1,  2,  3,  .  .  .  elements,  the  last  element  being  regarded  as  next  pre- 
ceding the  first. 

It  maybe  noted  that  (sQ~^  =  ^-^s~\  For  (st)~^  {st)=l,  and 
by  multiplying  the  members  of  this  equation  first  into  t^^  and  then 
into  s~\  we  obtain  the  result  stated. 

The  simplest  function  belonging  to  the  cycle  {xi  x^  .  .  .  x„)  is 

^  »X|  eX/2       I      •^2      3        i       •   •   '       I     *^)n 1  ^m       I     *^  w  *^1  • 

§  37.  Given  two  substitutions  Sa  and  Sp ,  if  we  wish  to  deter- 
mine the  group  of  lowest  order  which  contains  Sa  and  s^ ,  we  have 
not  only  to  form  all  the  powers  Sa^,  s^>^  and  to  multiply  these 
together,  but  we  must  form  all  the  combinations 

Of  the  substitutions  thus  formed  we  retain  those  which  are  dif- 
ferent from  one  another,  and  proceed  with  the  construction  until  all 
substitutions  which  arise  from  a  product  of  m  factors  are  contained 
among  the  preceding  ones.     For  then  every  product  of  m-\-l  fac- 


38  THEORY    OF    SUBSTITUTIONS. 

tors  is  obviously  reducible  to  one  of  m  factors,;  and  is  consequently 
also  contained  among  those  already  found.  The  group  is  then 
complete. 

In  case  s^Sa  =  SaSpf^,  the  con^esponding  group  is  exhausted  by  all 
the  substitutions  of  the  form  Sa^s^K     For  we  have  in  this  case 

S/s'  Sa  =  Sp-  Sa  Spf'  =  S|3  Sa  "  S^'^  =  §«  S^"", 
Sb   Sa=  Sb-  SaSB^'^  =  SaSfl^'^, 


S^'"Sa—  Sp"Sa,  •  Sa  =  SaSp"'^  Sa  =  Sa^Sp''^^'^, 
Sb"'sJ  =  Sa'sB"""^  Sa  =  Sa%'"''\ 


Consequently  any  product  of  three  factors  is  reducible  to  a  product 
of  two.     Thus 

S^PSa''S^-'  =  Sa''S^(-'  +  P^>''', 

and  the  theoreiji  is  proved. 
For  example,  let 

Si   —   \XlX2X^^X^X^) ,         §2  -—   V'^2'^3'^5^4/  ) 


then 


>1  —  yXiX2X^X^)  —  Sj  §2. 


The  group  of  lowest  order  which  contains  s^  and  So  contains 
therefore  at  the  most  5  •  4  =  20  substitutions.  To  determine  whether 
the  number  is  less  than  this,  we  examine  whether  it  is  possible  that 

If  this  were  the  case,  it  would  follow  that 

But  in  the  series  of  powers  of  S2  there  is  only  one  which  is  also  a  pow- 
er of  .9i ,  and  this  is  the  zero  power.  Consequently  we  must  have 
(i  =  yandft  =  d.  The  group  therefore  actually  contains  20  substi- 
tutions. These  are  the  following,  where  for  the  sake  of  simplicity 
we  write  only  the  indices: 


CORRELATION  OF  FUNCTIONS  AND  GROUPS.  39 

s,'  =  i,  s,  =  (2354),        s/  =  (25)  (34),        s,'  =  (2453), 

s^'  =  (1 2345),  s,s,  =  (1325),  s,s,'  =  (15)  (24),  s,s,'  =  (1435), 
si^=  (13524),  s,%  =  {l^S4:),  s,V=(14)(23),  s;%'  =  {12bi), 
s,'  =  (14253),  sX  =  (1243),  s,W  =  (13)  (45),  s,\'  =  (1523), 
si*  =  (15432),  si%  =  (1452),  s, V  =  (12)  (35),  s^ V  =  (1342). 
Analogous  results  may  be  obtained,  for  example,  for  the  case 

/»!  case  every  Sp^^Sa  (a^  =  1,  2,  3,  .  .  . )  can  be  reduced  to  the  form 
Sa^sp^,  the  group  of  lowest  order  which  contains  Sa  and  s^  is  exhausted 
by  the  substitutions  of  the  form  Sa''Sp\ 

For  by  processes  similar  to  those  above  we  can  bring  every  sub- 
stitution Sjs'^Sa*'  to  theforms/Sj3^.  The  proof  is  then  reduced  to  the 
preceding. 

Furthermore  if  s^"^  is  the  lowest  power  in  the  series  sp,  s^^^  .  .  . 
which  occurs  among  the  power  of  Sa?  then  the  group  contains  q  times 
as  many  substitutions  as  the  order  k  of  s^.  For  in  the  first  place,  if 
the  exponent  ^  in  s/s/  is  greater  than  q  —  1,  we  can  replace  s^g^  by 
a Sa'^sp" ,  where  v^q  —  1.  There  are  therefore  at  the  most  q  •  k differ- 
ent substitutions  s/S/g^.     Again  if 

then  we  must  have,  if  we  suppose  A  >  v, 

V~''  =  «a'*^''     (/^  — v<g— 1), 
and  consequently  X  —  v^[x  =  x..     There  are  therefore   actually  q  ■  k 
different  substitutions.     It  is  readily  seen  that  g  is  a  divisor  of  the 
order  r  of  s^. 

If  three  substitutions  Sa,  Sp,  Sy,  are  such  that  for  every  /j. 

S^''  Sa  —  Sj  Sp%    Sy'*  Sa  —  sj  S^^  Sy^ ,      Syl^  Sp  =  ««'  Sp"  Sy\ 

and  if  k  be  the  order  of  s^,  and  s^'^  the  lowest  power  of  Sp  ivhich  is 
equal  to  a  power  of  Sa,  s/,  finally  if  Sy  is  the  lowest  power  of  Sy 
which  is  equal  to  SaJ'Sp",  then  the  group  of  lowest  order  which  con- 
tains Sa,  sp,  Sy  is  of  order  kqt,  and  its  substitutions  are  of  the  form 

sJsp^Sy^   (^  =  0,  1,  .  .  .  Aj— 1;  e  =  0,  1,  . .  .  g— 1;  :'=  0,  1,  .  .  .  ^—1).. 

The  proof  is  simple  and  so  clearly  analogous  to  the  preceding 
ttiat  we  may  omit  it. 


40  THEORY    or    SUBSTITUTIONS. 

§  38.     With  this  set  of  propositions  belongs  also  the  following: 
If 

G  =  [l,  So,  S3,  ..  .s<] 

be  two  groups  of  substitutio7is  between  which  the  relation 

holds  for  all  values  of  a  and  ^,  and  if  furtheriyiore  G  and  H  have 
no  substitution  in  common  except  identity,  theyi  all  the  combinations 
Satp,  or  tpSay  (a  =  1,  2,  ...>;/?  =  1,  2,  ..  .  r')  form  a  group  of  order 
rr'  which  contains  G  and  H  as  subgroups. 

For  since 

Sat^  '  Syts—  Sa.(tpSy)ts  =  Sa(Se  '  t^)ts  =  S^ty, 

the  substitutions  §«  tp  form  a  group  of,  at  the  most,  rr'  substitutions. 
And  we  will  show  also  that  all  of  these  are  different  from  one  an- 
other.    For  if,  for  example, 

^a  f /3  -—  ^y  ts 

then  if  we  multiply  both  sides  of  this  equation  by  Sy~^  at  the  left 
and  t^"^  at  the  right  we  obtain 

Sy~    Sa=  t^t^~  . 

But  s^^^^a  is  a  substitution  of  G  and  t^t^^^  a  substitution  of  H,  and 
consequently  they  can  only  be  equal  if  both  are  equal  to  1.     Hence 

Sa  ^  ^7,  t^^^t^. 

The  substitutions  of  the  new  group  are  then  all  different,  and  the 
order  of  the  group  is  therefore  7t'.     We  denote  the  group  by 

K=\G,H\. 
We  add  without  proof,  the  following  generalization  of  the  last 
theorem. 

Under  the  same  assumption  Sat^  =  tySs,  if  the  two  groups  G  and 

rr^ 
H  have  A  substitutions  in  common  there  is  a  group  of  order  -^  which 

contains  G  and  A  as  subgroups.* 

§  39.     In  later  developments  a  group  will  frequently  be  required 

*F.  Giudice.    Palermo  Eeiid.  I,  pp.  222-223. 


CORRELATION  OF  FUNCTIONS  AND  GROUPS.  41 

the  order  of  which  is  a  power  of  a  prime  number  p.  The  exist- 
ence of  such  a  group  will  be  demonstrated  by  the  proof  of  the  fol- 
lowing proposition,  from  which  the  nature  of  the  group  will  also  be 
apparent. 

Theorem  XII.  If  p^  be  the  highest  power  of  the  prime 
number  p  ivhich  is  a  divisor  of  the  product  n!  —  1  •  2  •  3  .  .  .  n,  then 
there  is  a  group  of  degree  n  and  of  order  p. 

In  the  first  place  suppose  n  <  p\  so  that  n  =  ap  -\-b  {a,b<Cp). 
Then,  of  the  numbers  1,  2,  3,  .  .  .  w,  only  p,  2p,  3p,  ...  ap  are 
divisible  by  p,  so  that  f=  a.  We  select  now  from  the  n  elements 
a  systems  of  p  Zetters  each,  aud  form  from  each  system  a  cycle,  as 
follows : 

Then  the  group  which  arises  from  these  is  the  group  required : 

A-l  =  |_Si  ,  Si  ,   .  .  .  §2  ,§2  ,  .  .  .  S„ ,  S„  ,  .  .  .  J  =  I  Sj  ,  §2 ,  .  .  .  S„  [ . 

For  every  Sa  with  its  various  powers  forms  a  subgroup  of  order  p, 
and  since  no  two  of  these  a  subgroups  have  any  element  in  common, 
it  follows  from  Theorem  II  that 

S^PS^'^^S^'^S^P, 

Accordingly  every  possible  combination  of  substitutions  Si%  Sg^,  .  . . 
belonging  to  K  can  be  brought  to  the  form 

s.'^s/sJ  ....  s/     («,  ,3,  ^, . . .  V  =  0, 1,  2,  . . .  p— 1). 

The  group  K  therefore  contains  at  the  most  p"  substitutions.  And 
it  actually  contains  this  number,  for  all  these  p'*  operations  are  dif- 
ferent from  one  another.     For  if 

s,''s/s,y  .  .  .  s/  =  s^^'sfs^y  .  .  .  s/', 
it  would  follow  that 

s^-«'sj«  z=  Si''-'^'  —  s^^'s^y' . . .  s,/'s«-%-r'* .  •  'S^-ys^^^  =  s^f^^s^y^'y. . . 

and  therefore,  since  Sj ,  has  no  element  in  common  with  §2 ,  Sg ,  .  .  .  , 
we  must  have  a  —  a',  etc. 

Again,  if  n=p\  we   shall   have/  =  p  +  l,  since  in'^the  series 

*In  the  designation  of  a  group  the  brace,  j  i ,  as  distinguished  from  the  braclcet,  T  | « 
indicates  that  the  group  referred  to  is  the  smallest  group  which  contains  the  included 
substitutions.  The  bracket  contains  aMthe  substitutions  of  the  group  considered,  while 
the  brace  contains  only  the  generating  substitutions.  The  latter  can  generally  be  se- 
lected in  many  ways.    (Jf.  the  notation  at  the  close  of  the  last  Section. 

3a 


42  THEORY    or    SUBSTITUTIONS. 

1,  2,  3,  .  .  .p\  the  numbers  p,  2p,  3p,  . . .  (p  —  l)p,  p^  are  divisible 
by  p.     We  form  now  again  the  substitutions 

as  before, and  in  addition  to  these  the  substitutions^,^!  which  affects 
all  the  p^  elements 

S^,  +  1  —  yXi^X^-^Xi^  .  .  .  X^^Xi^X^.^  .  ,  .  XiJ)Ci.^  .  .  .  Xi^  .  .  .  Xt    \ 

Then  the  required  group  is 

For  in  the  first  place  we  can  readily  show  that 


^1    ^i^  +  1     ^^  ^P  +  1  ^2  ) 

*aSj,  +  i        —  *i^  +  l    *«+!» 

Sj,    *i>  +  1   =  ^p  +  1  *1  > 

*1   ^P  +  1      ^^  ^P  +  1*2   ) 

*a  *iJ  +  l      =S^>  +  1    Sa+i    , 

S^,    *iJ  +  1    ^^  ^p  +  1  *1     J 

*i  *jp + r  ^^  *i> + 1  *3 ) 

„A,„          2     —               2„          A 

*jj    *iJ  +  1    ^=  S^J  +  1    *2    > 

^1    Sp  +  i'*  =  Sp  +  i'*Sjx  +  i    J 

*a    Sp  +  i'*  =^  Sp  +  1^  Sa  ^  ^    , 

*iJ    *ZJ  +  l'*=S^>  +  l'^*i>  • 

Accordingly  every  combination  of  the  substitutions  Sj ,  Sa ,  .  .  .  s^,  +  j 
can  be  brought,  as  in  §  37,  to  the  form 

Si^^V  . . .  V^i^+i"    («j  /^j  r, . . .  ^  ^  =  0, 1,  2, . . .  p  — 1). 
But  we  must  also  show  in  the  present  case  that  we  need  only  take 
the  powers  of  s^  +  ^  as  far  as  the  (p — l)**^.     We  find  that 

^2>+l     ~^  K'^ii'^io'^is  •  •   •  "^i^))     Xp^ki^k^  •   •   •  "^k^,)  •   •   •   \p^ti'^t»  •   •   •  '^<„) 

=  Si  §2  .   .   .  Sj,  , 
i)  +  1        —  *1   *2     •  •  •  ^p  • 

Consequently,  if  A;>p,  we  can  replace  the  highest  power  of  s^,^i^' 
which  occurs  in  Sj^,j^^  by  powers  of  Sj,  Sg,  . .  .s^,,  and  these  can  then 
be  written  in  the  order  above. 

The  question  then  remains  whether  the  p^' +  ^  =p^  substitutions 
thus  obtained  are  all  distinct.     If  two  of  them  were  equal 

n  an  B  O  <■<}  " —  o  "■' o  /3'  o   t'o  f' 

bl    6.2      '   '   •  ^p^p  +  l     —  oi      <^2        '  '   '  f>p    Op  +  1      > 

we  should  have  ^ 

„  k  —  k'  —  Qa'  —  aoBf—B  q  J  —  t 

^P  +  i  —  ^1  ^-^         '^  .  .  .  Sj, 

But  the  substitution  on  the  right  does  not  affect  the  first  subscripts 
i,  k,  .  .  . ,  while  that  on  the  left  does,  unless  /.  =  /.'.  The  proof  then 
proceeds  as  before. 

If  n>p'^  but  <p'\  that  is,  if  n  =  ap'^ -\-bp  -{-  c   (a,  6,  c<p), 
we  select  from  the  n  elements  X\  any  a  systems  of  p^  elements  each, 


CORRELATION  OF  FUNCTIONS  AND  GROUPS.  43 

and  any  other  b  systems  of  p  elements  each.  With  the  former  we 
construct  a  groups  K2,  and  with  the  latter  b  groups  K^ .  The  com- 
bination of  these  a-\-b  groups  gives  the  required  group  K^.  For 
the  product  of  the  numbers 

{a-l)p'  +  l,  (a-l)p«  +  2,  .  . .  {a-l)p^+p 

{a-l)p^+p\     (a<p) 

is  divisible  by  only  the  same  power  of  p  as  the  product  of 
l,2,...p,...p\ 
Again,  if  n  =p^,  then  the  exponent  /  of  p^  is  increased  by  1  on 
account  of  the  last  term  of  the  series 

(p-i)p'+i,{p-i)p'+^,  ■  ■  ■  (p-'i-)p'+p,  ■ .  ■ 
(p-i)p'+p'-^p\ 

so  that  in  this  case  the  multiplication  of  the  p  partial  groups  K2  is 
not  sufficient.  In  this  case,  exactly  as  forn=:p^,  we  add  another 
substitution  which  contains  all  the  p  ■  p^  elements  in  a  single  cycle, 
and  the  p^^^  power  of  which  breaks  up  into  the  p  substitutions  as  inr 
the  case  of  s^,  +  1  above.  Then,  as  in  that  case,  we  can  show  that  the 
new  group  satisfies  all  the  requirements.  At  the  same  time  it  is 
clear  that  the  method  here  followed  is  perfectly  general,  and  accord- 
ingly the  theorem  at  the  beginning  of  the  Section  is  proved. 

§  40.     Since  all  the  groups  K^^  K^^  K^,  .  .  .  enter  into  the  forma- 
tion of  the  group  K,  we  have  the  following 

Corollary.     If  p^  is  the  highest  power  of  p  which  is  a  divi- 
sor of  n\ ,  {hen  we  can  construct  a  series  of  groups  of  n  elements 

1,  i^i ,  K2 ,  .  .  .  K\ ,  K\  +  1 ,  ...  Kf 

ivhich  are  of  order  respectively 

l,p,p',  ...p^p^  +  ^  ...p-^. 

Every  group  K^  is  contained  as  a  subgroup  in  the  next  following 


CHAPTER  IIL 


THE    DIFFERENT   VALUES   OF    A    MULTIPLE-VALUED    FUNC- 
TION AND  THEIR  ALGEBRAIC  RELATION 
TO  ONE  ANOTHER. 

§  41.  We  have  shown  in  the  preceding  Chapter,  that  to  every 
function  of  n  elements  x-^,  X2,  x^^  .  .  .  x„  there  belongs  a  group  of 
substitutions,  and  that  conversely  to  every  group  of  substitutions 
there  correspond  an  infinite  number  of  functions  of  the  elements. 
The  examination  of  the  relations  between  different  functions  which 
belong  to  the  same  group  we  reserve  for  a  later  Chapter.  The 
problem  which  we  have  first  to  consider  is  the  determination  of  the 
connection  between  the  several  values  of  a  multiple -valued  function 
and  the  algebraic  relations  of  these  values  to  one  another. 

If  9?  (i^i ,  a?2 ,  .  .  .  x^  is  not  a  symmetric  function,  or,  in  other 
words,  if  the  substitutions  Si  =  1,  S2,  Sg,  .  .  ..s,.  of  the  group  G  be- 
longing to  (f  do  not  exhaust  all  the  possible  substitutions  (^.  e.  r  <  w!), 
then  (f^  on  being  operated  upon  by  any  one  of  the  remaining  substi- 
tutions <r2,  will  take  a  new  value  ^2.=  ^<ro- 

We  proceed  to  construct  a  table,  the  first  line  of  which  consists 
of  the  various  substitutions  of  the  group  G:  • 

§1  =  1,  82,83,  . . .«,-;    G',    ipy. 

The  second  line  is  obtained  from  the  first  by  right  hand  multi- 
plication of  all  the  substitutions  Sa  by  a^ .     This  gives  us 

^2,    82<^21   ^Z^1'>   •  •  •    ^r'^2;         G  ■   <72;        ^2' 

We  show  then,  as  in  Chapter  II,  §  35,  1)  that  all  substitutions  of 
this  line  convert  ^j  into  ^2?  for  since  ^^^  =  ^1,  it  follows  that 
<P^^a^=-  9<T^  —  <p2'i  2)  that  no  other  substitutions  except  those  of  this 
line  convert  (p^  into  02;  for  if  r  is  a  substitution  which  has  this 
effect,  then  we  shall  have 


MULTIPLE-VALUED    FUNCTIONS ALGEBRAIC    RELATIONS.  45 

SO  that  r<72""^  leaves  the  function  ^i  unchanged;  consequently 
rr72~^  =  Sx  and  t  =  (t(T2~^)(72  =  s^fT^^,  aud  therefore  r  is  contained  in 
the  second  line;  3)  that  all  substitutions  of  this  line  are  distinct;  for 
if  Sa(r2  =  8^(70,  it  follows  that  Sa  —  Sp-,  4)  that  the  substitutions  of 
this  line  are  all  different  from  those  of  the  first;  for  the  latter  all 
leave  <Pi  unchanged,  w^hile  the  former  all  convert  (p^  into  ^2- 

If  the  2r  substitutions  Sa  and  s^^.^  do  not  yet  exhaust  all  the  pos- 
sible n\  substitutions,  then  any  remaining  substitution  0-3  will  con- 
vert (pi  into  a  new  function  <p^^  =  ^3 ;  for  all  the  substitutions  which 
produce  <Pi  and  ^2  are  already  contained  in  the  first  two  lines.  By 
the  aid  of  ^73  we  form  the  third  line  of  our  table 

The  substitutions  of  this  line  have  again  the  four  properties  just 
discussed.  They  are  all  the  substitutions  and  the  only  ones  that 
convert  9^1  into  (p^ ,  and  they  are  all  different  from  one  another  and 
from  those  of  the  preceding  lines. 

If  these  3r  substitutions  do  not  exhaust  all  the  possible  n!,  we 
proceed  in  the  same  way,  until  finally  all  the  nl  substitutions  are 
arranged  in  lines  containing  r  each. 

We  shall  frequently  have  occasion  to  construct  tables  of  this 
kind.  All  these  tables  will  possess  the  properties:  1)  that  all  the 
substitutions  in  any  line  will  have  a  special  character;  2)  that  only 
the  substitutions  of  this  line  will  have  this  character;  3)  that  all  the 
substitutions  of  any  line  are  different  from  one  another;  and  fre- 
quently, but  not  always,  the  fourth  property  will  also  appear:  4) 
that  all  the  substitutions  of  any  line  are  different  from  those  of  any 
other  line.  * 

Summarizing  the  preceding  results  we  have  the  following 

Tlieorem  I.  If  the  multiple-valued  function  <p{xi,X2,  .  .  .  x„) 
has  in  all  p  values  ^1,  ^25  ^sr-  •  •  S^p  <^**^  ^f  9  *^  converted  successively 
into  these  values  by  certain  substitutions,  for  example  Ij  fl^a,  rt"., .  .  .  ^-p, 
furthermore,  if  G,  the  group  of  (p,  is  of  order  r  and  coritains  the 
substitutions  Si  =  1,  S2,  s^,  .  .  .s,.,  we  can  arraiige  all  the  possible  n\ 
substitutions  as  in  the  following  table : 

*Such  tables  were  given  by  Cauchy:    Exercices  d'analyse  et  de  physique  matliemat- 
ique,  III.,  p.  184. 


46  THEOEY    OF    SUBSTITUTIONS. 

(p2\  <J"2,      SatTg,     S3  (72)   •  .  •  8r<^l\  G\  ' 

V'a;       'J-g,    SgO-g,   S3<73,  .  .  .  s,.<r3;        (^^ . 


(pp\  (Tp,      S2<Tp,    S^ffp,  .  .  .  Sr<Tp]  Gj  •  <Tp 

in  which  every  line  contains  all  and  only  those  substitutions  which 
convert  (f  into  the  value  <Pa  prefixed  to  the  line. 

The  several  values  ^1 ,  s^2  >  •  •  •  S^p  of  the  function  <p  are  called  con- 
jugate values. 

§  42.  From  the  circumstance  that  all  the  substitutions  of  this 
table  are  difiPerent  from  one  another,  and  that  the  p  lines  of  the 
table  exhaust  all  the  possible  substitutions  we  deduce  the  following 
theorems : 

Theorem  II.  The  order  r  of  a  group  G  of  n  elements  is  a 
divisor  of  n ! 

Theorem  III,  The  number  p  of  the  values  of  an  integral 
function  of  n  elements  is  a  divisor  of  n\ 

Theorem  IV,  The  product  of  the  number  p  of  the  values 
of  an  integral  function  by  the  order  r  of  the  corresponding  group 
is  equal  to  n\ 

The  third  theorem  imposes  a  considerable  limitation  on  the  pos- 
sible number  of  values  of  a  multiple  valued  function.  Thus,  for 
example,  there  can  be  no  seven-  or  nine-valued  functions  of  five 
elements.  But  the  limits  thus  obtained  are  still  far  too  great,  as  the 
investigations  of  Chapter  V  will  show. 

§  43.  Precisely  the  same  method  as  that  of  §  41  can  be  applied 
to  the  more  general  case  where  all  the  substitutions  of  the  group  G 
belonging  to  <p  are  contained  in  a  group  H  belonging  to  another 
function  4',  so  that  G  is  a  part  or  subgroup  of  H,  just  as  in  the 
special  case  above  G  was  a  subgroup  of  the  entire  or  symmetric 
group.  We  see  at  once  that  all  the  substitutions  of  H  can  be 
arranged  in  a  series  of  lines,  each  line  containing  r  substitutions  of 
the  form  Sa'J'm  (A  =  1,  2,  . .  .  r).  And  we  pass  directly  from  the  pre- 
ceding to  the  present  case  by  reading  everywhere  for  "all  possible 
substitutions"  simply  "all  the  r^  substitutions  of  W\  We  have 
then 


MULTIPLE -VALUED  FUNCTIONS ALGEBRAIC  RELATIONS.       47 

Theorem  V.  If  all  the  r  substitutions  of  the  group  G  are 
contained  among  those  of  a  group  H  of  order  r^ ,  then  r  is  a  divisor 
of  r^. 

Theorem    VI.       Given  two  functions  <p  {x^^x^,  ...  x,)  and 
0  (xi,  a^2)  •  •  •  ^w))  if  4'  retains  the  same  values  for  all  substitutions 
which  leave  (p  unchanged^  the  total  number  of  values  p  of  (p  is  a  mul- 
tiple of  the  total  number  of  values  />,  of  (p. 
For  we  have 

_n\  _n\  P  _  fi 

^  ~  r  r^^    '    '   p^~  r' 

Corollary.  If  a  function  <p{x^,  ...  a'„)  belongs  to  a  subgroup 
G  of  the  group  H,  and  if  r  is  the  order  of  G  and  r^  that  of  H, 
then  ip  on  being  operated  upon  by  the  substitutions  of  H  takes  exactly 

—  values, 
r 

§  44.     By  a  still  further  extension  of  the  subject  we  may  include 
the  case  where  two  groups   G    and  H   contain  any   substitutions 
in  common.     This  case  can  be  at  once  reduced  to  that  of  the  prece- 
ding Section.     For  this  purpose  we  employ  the  following  proposi 
tion : 

Theorem  VII.  The  substitutions  common  to  two  groups 
form  a  new  group,  the  order  of  which  is  accordingly  a  divisor  of  the 
orders  of  both  the  given  groups. 

For  if  (T  and  r  belong  to  both  G^  and  6r2  then  <r .  r  also  belongs  to 
both  Gi  and  G.2,  and  occurs  among  the  common  substitutions.  The 
same  result  can  also  be  obtained  as  follows.  If  (f^  and  (P2  be  func- 
tions belonging  to  G^i  and  G.,  respectively,  then  the  function 

where  a  and  /5  are  arbitrary  constants,  remains  unchanged  for  those 
substitutions  which  leave  both  ^1  and  ^2  unchanged,  that  is,  which 
are  common  to  G^  and  G^.  These,  being  all  the  substitutions  which 
belong  to  ^'',  form  a  group. 

Corollary  I.  Two  groups  whose  orders  are  prime  to  each 
other  can  have  710  substitutions  in  common  except  the  identical  sub- 
stitution. 


48  THEOEY    OF    SUBSTITUTI0N8. 

Corollary  II.  The  order  of  every  group  H  iddch  consists  of 
all  or  a  part  of  the  substitutions  common  to  the  ttvo  groups  G^  and 
G2  is  a  divisor  of  both  r^  and  r. 

§  45.     We  proceed  next  to  determine  and  tabulate  the  groups 
which  belong  to  the  various  values  v'/  of  s^,  (^  =;  1,  2,  3,  .  .  .  p). 
The  group  G—G^oi  (fi  contained  the  substitutions 

Sj  =  1,  §2)  ^3j    •   •   •   ^c 

The  value  ^2  of  <p  was  obtained  from  <p^  by  the  substitution  (t.,  ;  con- 
sequently a~f  ^  converts  (f^  back  into  <Pi .  If  then  we  apply  to  tp^  suc- 
cessively the  substitutions  ff2~^,  ««?  <^2?  the  first  of  these  will  convert 
^2  into  (pi ,  the  second  will  leave  (f^  unchanged,  and  the  third  will  con- 
vert ^1  back  into  (fo.  It  appears  therefore  that  ff^  is  unchanged  by 
every  substitution  of  the  form  <5*2~^Sa^2j  («  =  1,  2,  .  .  .  r).  For  the 
second  line  of  our  table  we  take  therefore 

ff-^S^a^—l,         (T2~^S2<r2j         ^2~^S3'^2»      •••     <^2~^S.<^2- 

We  can  then  show  that  this  line  contains  all  the  substitutions 
belonging  to  the  group  of  ff^.  For  if  r  be  any  substitution  which 
leaves  fa  unchanged,  then  ra^^  will  convert  9^2  hito  fi,  that  is 

Consequently  the  substitution  <72'^'^2'~  ^  belongs  to  the  group  of 
Vi,  and  we  may  write  it  equal  to  ««.  But  from  the  equation 
(j^raf'^^=^Sa  follows  r  —  <j.r^{<T.2'<J'r^)^2~^2~^Sa'^2^  ^s  was  asserted. 
Again  it  is  easily  seen  that  all  the  substitutions  of  the  second 
line  are  different  from  each  other.     For  if 


S„^(T.> 


it  follows  that 


We  may  note  however  that  the  substitutions  of  the  second  line 
are  not  necessarily  different  from  those  of  the  first.  In  fact  the 
identical  substitution  is  of  course  always  common  to  both  and  other 
substitutions  may  also  occur  in  common.     {Cf.  §  50). 

From  the  three  properties  of  the  second  line  obtained  above,  it 
follows  that  the  r  substitutions  of  this  line  form  the  group  of  (p^. 
We  will  denote  this  group  by  G^,.  That  these  substitutions  from  a 
group  can  also  be  shown  formally;  for 


MULTIPLE-VALUED    FUNCTIONS ALGEBRAIC    RELATIONS.  49 

{(TrT  ^80.(^2)  {^2~  ^8^  (7.2)  =  (7-2"  ^Sa  (^2'^2~  ')  ^/S  ^2  =   <^2'\SaS^)  ^2  , 

SO  that,  if  Sa?  s^»  .  .  •  form  a  group,  as  was  assumed,  the  same  is  true 
of  the  new  substitutions. 

Similar  results  hold  for  all  the  other  values,  <p3,  <Pi,  .  .  .  <pp  oi  <p, 
and  we  have  therefore 

Theorem  VIII.     //  the  values  <Pi,  <p2,  .  -  -  <Pp  of  a  p-valued 
function  <p  proceed  from  (p  by  the  application  of  the  substitutions 

ff^  — 1,0-2,  'J-g,  .  .  .  (7p,  then   the  groups  G^,  G2,  .  .  .  Gp  of  <Pi,  <p2<,  .  .  .  <Pp 
respectively  are 

G,  =  \_s,  =  l,S2,S3,  .  .  .s,], 

^2  =  1^2"  ^  Sl'^2  =  1 5    '^r  ^  *2'^2  J    ^2~  ^  Sg^Tg  ,   .  .  .  (72~  ^  S,  (T,,]     =  fT.^-^  G^  (T2  „ 
^P  —  ['^p""^«l'^P  =  I5   '^P~'«2'yp,   ^^p'^S^rTp,  .  .  .  <7p-^S,rTp\  =  ^Tp-'G^fTp^ 

§  46.  The  functions  (fu  (P2,  -  •  •  fp  are  of  precisely  the  same^ 
form  and  only  differ  in  the  order  of  arrangement  of  the  x^^s  which 
enter  into  them.  Such  functions  we  have  called  (§  3)  similar  or  of 
the  same  type.  Accordingly  the  corresponding  groups  Gi,  G2,  ... 
must  also  be  similar  or  of  the  same  type,  that  is  they  produce  the 
same  system  of  rearrangement  of  the  elements  X)^  and  only  differ 
in  the  order  in  which  the  elements  are  numbered.  This  is  clear  d 
priori,  but  we  can  also  prove  it  from  the  manner  of  derivation  of 
^r  ^Sa'^i  from  Sa ,  and  in  fact  we  can  show  that  not  only  the  groups 
Gi,  G2,  .  .  .,  but  also  the  individual  substitutions  Sa  and  (^r^s^^i  are 
similar;  that  is,  these  two  substitutions  have  the  same  number  of 
cycles,  each  containing  the  same  number  of  elements.  The  process 
of  deriving  the  substitutions  (rr^Sa'^i  from  the  SaS  is  called  trans- 
formation. The  substitution  trr-  ^s^^r,  is  called  the  conjugate  of  Sa 
with  respect  to  «t„  and  similarly  the  group  Gi  is  the  conjugate  of 
Gi  with  respect  to  o-,.  We  shall  denote  this  latter  relation  frequently, 
as  above,  by  the  equation 

To  prove  the  similarity  of  s^  and  o--^s^/7;,  let  us  suppose   that 
(a?i ,  a?2 ,  .  .  .  Xa)  is  any  one  of  the  cycles  of  s^  and  that  (t^  replaces 
x^,  X2,  .  .  .Xahj  Xi^,  Xi,,  .  .  .  Xi^y  so  that,  in   the  notation  A  of  §  22, 
<7  may  be  written 
4 


50  THEORY    OF    SUBSTITUTIONS. 


I   ^1    ^2     •  •   •    '^a   I 


Then  the  factors  of  the  substitution  (^r^Sa^^i  will  replace  x,-^  success- 
ively by  Xi,x.2j  08i.,',  similarly  x,-,  will  be  replaced  by  x,-.^^  x,-^  by  X/^, 
and  finally  x,-^  by  x,-^.  Accordingly  ^r^Sa^,-  contains  the  cycle 
(a?,j  a?^  .  .  .  XiJ,  and  this  is  obtained  from  the  corresponding  cycle 
{xi  a?2 .  .  .  'X^a)  of  Sa  by  regarding  this  cycle,  so  to  speak,  as  a  function 
of  the  elements  x^  and  applying  to  it  the  substitution  (t,.  In  the 
same  way  every  cycle  of  <yr^Sa<*i  proceeds  from  the  corresponding 
cycle  of  Sa.  The  two  substitutions  have  therefore  the  same  number 
of  cycles,  each  containing  the  same  number  of  elements,  as  was  to 
be  proved. 

Example:     The  function  of  four  elements 

(pl  —  X^  X2  ~\~  '^3  ^i 

has,  as  we  have  seen,  three  values,  and  its  group  G^  is  of  order 
-^  =  8.  The  substitutions  (T2  =  (^2^3)  and  ^g  =  {XiX^},  which  are  not 
contained  in  G^ ,  convert  (f^  into 

<f2  ^^=  XiX^-f-  X2X^, 

Cg  ^=^  X^X^-f-  X2X^.  . 

respectively.  By  transposition  with  respect  to  ^o  and  (t.^,  we  obtain 
from 

Ctj  =  [_l,(£CiX2),   {X^X^jj  {X1X2)  (.^3^4),  (XiX^)  \X2'^i),  (^1.^4)  {X2X^)f  {^XiX-^X2X^) 

the  two  groups  belonging  respectively  to  ^2  and  fg , 

G2  =  (^2"   Gri^2  ^^ 

j_l,  (^^liCgj,    (^i3?2^4J,    (^1^3/  \'^2'^4/)    \p^l'^2)  [p^S'^iJj    X'^l'^i)  \^2'^3)i    (^CCj^O'^S'^-tjj 
\XiX^XrfiC2)  J  , 

G3  =  (Tf~    Gi<Ts  = 

ll,{XiX^),  (X2Xs),  {XiXi){X2Xs),  {x,Xs)(X2X^),  (x,X2){x,x,),  {X^X^X,X2), 
{XiX2X^X^)^. 

§  47.  Corollar J'  I.  //  a  group  of  substihdions  is  trans- 
formed with  respect  to  any  substitution  ivhatever,  the  transformed 
substitutions  form  a  group. 

Corollary  II.  The  two,  generally  different,  substitutions 
SaSp  and  SpSa  are  similar.     For  SaSp  —Sp~^{spSa)sp.. 


MULTIPLE-VALUED    FUNCTIONS ALGEBRAIC    RELATIONS.  51 

Corollary  III,  The  substitution  s^s^s^^  is  conjugate  to  s^ 
with  respect  to  Sa"^- 

Corollary  IV.  If  the  substitution  Sa  is  of  order  r  and  if  s^ 
be  such  that  its  q^^  and  no  lower  power  occurs  among  the  powers  of 
Sa ,  and  if  furthermore  the  conjugate  of  s^  with  respect  to  Sa  is  a 
power  of  Sp,  then  the  smallest  group  containing  s^  and  s^  is  of  order 
q  r.  {cf  §§  37,  38,  Chapter  II). 

Corollary  V.  All  substitidions  which  transform  a  given 
substitution  s  into  its  powers,  s*,  form  a  group. 

Corollary  VI.  All  substitutions  which  transform  a  given 
group  into  itself  form  a  group. 

Corollary  VII.  If  tivo  substitutions,  or  two  groups,  are 
similar,  substitutions  can  alicays  be  found  tchich  transform  the  one 
into  the  other.  In  the  case  of  two  substitidions  the  transforming 
substitution  is  found  at  once  from,  §  46.  In  the  case  of  groups,  ive 
have  only  to  construct  the  corresponding  functions  and  determine 
the  substitution  a^  which  converts  the  one  into  the  other. 

Corollary  VIII.  Ttvo  powers  s"-  and  s^  of  the  same  substi- 
tution are  similar  when,  and  only  when,  a  and  /?  have  the  same 
greatest  common  divisor  ivith  the  order  of  s. 

§  47.  We  turn  now  to  a  series  of  developments  relating  to  the 
existence  of  certain  special  types  of  groups  analogous  to  those  of 
§  39,  Chapter  II. 

Given  any  group  G  of  order  g,  let  H^  of  order  h^  and  K^  of  order 
ki  be  subgroups  of  G,  and  let  (f^  and  v''i  be  functions  belonging  to  H^ 
and  Ki  respectively.  These  functions,  on  being  operated  on  hj  all 
the  substitutions  of  G,  take  respectively  ~=  /ioand-|-  =  k^  values 
in  all  (§  43,  Corollary) ;  let  these  be  denoted  by 

<Pi,  <P2,"  '  <Ph,   and   (^'i,  0,,  .  .  .  ^V^. 

The  group  of  any  one  of  the  functions  c''^  will  be  Ka=  (^a'^Kiffa, 
where  ^a  is  any  substitution  of  G  which  converts  (J\  into  (/'a. 
We  form  now  the  entire  system  of  values 

a  9-x  +  ^  V'V      i^'  =  1,  2,  .  .  .  ho;  /j.  =  1,  2,  .  .  .  k^), 


52  THEORY    OF    SUBSTITUTIONS. 

where  a  and  h  are  arbitrary  parameters,  and  divide  these  — ^  fnnc- 

1    1 
tions  into  classes,  such  that  all  the  functions  of  each  class  proceed 

from  any  one  among  them  by  the  operations  of  G. 

If  a(Px-\-hipa.  be  one  of  the  values  above,  and  if  we  apply  to  it 
all  the  substitutions  of  (r,  the  resulting  values  are  not  necessarily  all 
distinct.  In  particular  some  of  them  may  coincide  with  the  given 
value.  The  number  of  the  latter  is  equal  to  the  number  of  substi- 
tutions common  to  H^  and  -£"„=  '^oT^K^rr^.  Let  this  number  be  d^. 
Then  all  the  g  values  arising  from  a(p^-\-b  4' a  will  coincide  in  sets  of 
da  each,  as   is  easily  seen.     The  number  of    distinct  values   thus 

obtained  is  therefore  — - . 

da 

If  these  do  not  exhaust  all  the  possible  values  aifx-\-h  v'v,  let 
a  ^o-  +  6  ^T  be  any  remaining  value.  Then  in  the  same  class  with 
this  belongs  also 

From  the  latter  value  we  can,  as  before,  deduce  a  class  contain- 
ing in  this  case—  distinct  values,  where  d^  is  the  number  of  substi- 
tutions  common  to  H^  and  K^  —  (t^-  ^K^  (j^  . 

Proceeding  in  this  way,  we  must  finally  exhaust  all  the  -— —  val- 

ues  of  the  functions  a  s^a  +  6  v'v-  Writing  then  the  two  numbers  of 
values  equal  to  each  other,  we  have,  after  dividing  through  by  g , 

^  h,k,       d,^d,^        ^  dj 

where  m  denotes  the  number  of  classes  of  values  of  a(p\-{-h  4'^  with 
respect  to  the  group  G.      Since  /ij  is  a  multiple  of  every  da,  we  may 

write  —  =fa,  and  consequently 

(la 

B)  |-=/a  4-/2+   ...+/., 

where  the/'s  are  all  integers.  * 

§  49.  From  Theorem  V  of  §  43  it  follows  as  a  special  case  that 
if  a  group  contains  a  substitution  of  prime  order  p,  the  order  r  of 

♦Formulas  .4)  and  B)  were  obtained  by  G.  Frobenius,  Crelle  CI.  p.  281,  as  an 
extension  of  a  result  given  by  the  Author,  Math.  Annalen  XII F. 


MULTIPLE -VALUED    FUNCTIONS ALGEBRAIC    RELATIONS.  53 

the  group  is  a  multiple  of  p,  and  if  a  group  contains  a  subgroup  of 
order  p*  where  p  is  a  prime  number,  the  order  of  the  group  is  a  mul- 
tiple of  p*.  By  the  aid  of  the  results  of  the  preceding  Section  we 
can  now  also  prove  the  converse  proposition: 

Theorem  IX.  If  p*^  he  the  highest  power  of  the  prime  num- 
ber p  which  is  a  divisor  of  the  order  h  of  a  group  H,  then  H  con- 
tains subgroups  of  order  p*".  * 

In  the  demonstration  we  take  for  the  G  of  the  preceding  Sec- 
tion the  symmetric  group,  so  that  g  =  n\.  For  H^  we  take  the  pres- 
ent group  H,  and  for  K^  the  group  of  order  p^  of  §  39,  Chapter  II, 
p^ being  the  highest  power  of  p  which  is  a  divisor  oi  n\.  The 
formula  B)  of  §  48  then  becomes 
n ' 

^  =/,+/. +  ...+/„. 

The  left  member  of  this  equation  is  no  longer  divisible  byp;  con- 
sequently there  must  be  at  least  one  fp  =  -r-  which  is  also  not  di- 

visible  byp;  that  is  d^  is  divisible  byp*,  and  therefore  H  and  K^, 
the  latter  being  a  conjugate  of  K,  have  exactly  p*  substitutions  in 
cotnmon.  f     These  form  the  required  subgroup  of  H. 

Corollary.  At  the  same  time  it  appears  that  the  group  K  con- 
tains among  its  subgroups  every  type  of  groups  of  order  p^.  For 
we  need  only  take  any  group  of  order  py  for  H  in  the  above  demon- 
stration. 

§  50.     The  last  theorem  admits  of  the  following  extension : 

Theorem  X.  If  the  order  h  of  a  group  H  is  divisible  by 
p^,  then  H  contains  subgroups  of  order  p^. 

The  proof  follows  at  once  from  Theorem  XI,  as  soon  as  we  have 
proved 

Theorem  XI.  Every  group  H  of  order  p"^  contains  a  sub- 
group of  order  p"-~^. 

The  corollary  of  the  preceding  Section  permits  us  to  limit  the 

♦Cauchy,  loc.  cit,  proved  this  theorem  for  the  case  a  =  I.  The  extension  to  the 
case  of  any  a  was  given  by  L.  Sylow,  Math.  Annalen  V,  pp.  584-594. 

t  For  every  subgroup  of  K,  and  consequently  every  subgroup  of  K^,  has  for  its 
order  a  power  of  p. 


54  THEOEY    OF    SUBSTITUTIONS. 

proof  to  the  case  of  groups  of  order  p'^  which  occur  as  subgroups  in 
the  group  of  §  39,  Chapter  II.  The  group  K  =  Kf  there  obtained 
was  constructed  by  the  aid  of  a  series  of  subgroups  (§  40) 

1,  Ki,K2,  .  .  .  Kx,  ■K'a  +  i,  .  .  .  -^/-i, 
of  orders 

1,  p\  p\  ...  p\  p^  +  \  ..  .  2^^~\ 
every  one  of  which  is  contained  in  the  following  one.  If  the  group  H 
occurs  in  this  series,  the  theorem  is  already  proved;  if  not,  then  let 
Kx-^i  be  the  lowest  group  which  still  contains  H.  K^  then  does  not 
contain  all  the  substitutions  of  H.  We  apply  now  the  formula  B) 
of  §  48  to  the  groups  -STa  + 1 ,  K^,  and  H,  taking  these  in  the  place  of 
G,K,,SindH.     We  find 

P=/l+/2+    ...    +/»• 

This  equation  has  two  solutions,  since  the/'s,  being  divisors  of 
h=p'^,  are  powers  of  p:  either  we  must  have  fi=f2—  ...  =1  and 
m  =p,  or  else  m  =  l  and  /i  =  p.  In  the  former  case  it  would  follow 
that  h  =di,  i.  e.,  that  iJ  is  a  subgroup  of  Kx,  which  is  contrary  to 
hypothesis.  Consequently /i  =  p,  i.  e.  H  and  Kx  have  a  common  sub- 
group of  order  p*~\     This  is  the  required  group.  * 

To  Theorem  X  we  can  now  add  the  following 

Corollary.  Every  group  of  order  p«  .  pi*i  -pa*^  •  .  .  cctw  he  con- 
structed by  the  combination  of  subgroups,  one  of  each  of  the  orders 

A  smaller  number  of  subgroups  is  of  course  generally  sufficient. 

A  further  extension  of  the  theory  in  this  direction  is  not  to  be 
anticipated.  Thus,  for  example,  the  alternating  group  of  four  ele- 
ments, which  is  composed  of  the  twelve  substitutions 

1,   {X1X2)  [X'^X^),   {XiX^)  [X2X^),   [X^X^)  {XoX-^J, 
{XiX2X^j,        {XiX2Xi)y        [XiX^X^j^        \X2X'^X^j,       ^ 
\XiX^X2),       {XiX^XoJ,        [XiX^^Jj        \p^2'^i'^s)} 

has  no  subgroup  of  order  6. 

§  51.  We  insert  here  another  investigation  based  on  the  con- 
struction of  tables  as  in  §  41. 

Let  if  be  a  group  of  order  h  affecting  the  n  elements  x^,  X2,  .  .  .  x„. 

*  G.  Frobenius:  Crelle  CI.  p.  383-4. 


MULTIPLE- VALUED    FUNCTIONS ALGEBEAIC    RELATIONS.  55 

From  these  n  elements  we  arbitrarily  select  any  fc,  as  o^j,  0^2,  .  . .  x^, 
and  let  H'  be  the  subgroup  of  H  which  contains  all  the  substitu- 
tions of  the  latter  that  do  not  afPect  x-^,  X2,  .  .  .  Xj,.  Suppose  h' 
to  be  the  order  of  H',  and  ^1  =  1  ^2  ?  •  •  •  h'  to  be  its  several  substitu- 
tions. We  proceed  then  to  tabulate  the  substitutions  of  H  as  fol- 
lows: 

Given  any  substitution  8a  of  H,  suppose  that  this  converts 
x^,x.2,  .  .  .Xj,  into  Xa^  ,Xa,...  Xaj, ,  iu  the  order  as  written.  Then  all 
the  substitutions 

also  convert  Xi,X2,  ...  Xk  intoa^a^,  cCa.,,  •  •  •  ^a^  respectively,  and  these 
are  the  only  substitutions  of  H  which  have  this  effect.  We  take 
these  various  sets  of  h  substitutions  for  the  lines  of  our  table,  which 
is  accordingly  of  the  form 

1,       C2j  h)  •  '  •      ^/i'  J 

§2  5      ^2^2  ?      ^3^2 )       •   •  •       ^7i'  ^2 »  ** 

1  J  S3 ,      Co^a )       ^3^3  J       •  •   •      ^/j'  ^3 ) 


The  substitutions  of  the  table  are  obviously  all  different,  and  conse- 
quently fj.  h'  =  h. 

Again,  suppose  that  v^  is  any  substitution  of  H  which  contains 
among  its  cycles  one  of  order  k,  say 

(1)  OJ^  =  {X^X^X^  .  .  .  Xj,). 

Then  all  the  (necessarily  distinct)  substitutions 

(2)  V,,t2V^,t^Vy^    .    ..th'Vi 

will  contain  the  same  cycle  (1)  and  these  will  be  the  only  substitu- 
tions of  H  which  contain  this  cycle.  We  wish  now  to  determine  how 
many  substitutions  of  H  contain  either  the  cycle  (1)  or  any  cycle 
obtainable  from  (1)  by  transformation  with  respect  to  the  substitu- 
tions of  H,  say 

\^a)  ^a^^  {Xa^Xa.2Xa^  .  .  .  ^a^)' 

Now  since  all  the  substitutions  of  the  same  line  of  T)  convert 
the  elements  of  the  cycle  (1)  into  one  and  the  same  system  of  ele 
ments,  it  follows  that  if  we  write 


56  THEORY    OF    SUBSTITUTIONS. 

Sa~'ViSa  ~Va        («  =  1,  2,    .   .   .   fj), 

where  each  Sa  is  the  same  as  that  of  the  table  T),  then  all  the  conju- 
gates of  the  cycle  (1)  which  occur  in  the  substitutions  of  H  are  con- 
tained in  Vj ,  ^2  ?  •  •  •  ^/x  •  Suppose  the  notation  so  chosen  that  o)^  is 
contained  in  Va.  If  now  we  denote  the  substitutions  of  H  which  do 
not  affect  Xai,Xa^_,  .  .  .  a?a^.,  i.  e.  those  of  the  group  Sa~^H'Sa  =  Ha  by 

1      f  (a)     f  (a)  f'ia) 

and  by  right  hand  multiplication  by  Va  form  the  line 

(3)  Va,    t^^'^K,    ts^'^ha,     ...    tj,'Va, 

then  (3)  contains  all  the  substitutions  of  H  which  involve  the  cycle 
(Oa.     The  /-/  lines  of  the  following  table 

^2,      U^^V^,       U^^V.2,       ...      tj/'^V„ 

therefore  contain  all  the  required  substitutions. 

The  question  then  arises,  how  many  of  the  lines  of  To)  give  the 
same  cycle;  for  example,  in  how  many  lines  the  cycle  (1)  occurs.  If 
this  cycle  occurs  in  i?^  =  s^"  ^^i^t  ?  then  Sr  must  permute  the  elements 
x^,X2,  .  .  .Xj,  cyclically,  and  must  therefore  contain  a  power  of  (1) 
as  a  cycle.  Consequently  we  must  have  Sr  equal  to  one  of  the  sub- 
stitutions Vi,  Vi^fo  ...  ^^i*""^  or  one  of  these  multiplied  by  some  U. 
But  the  substitutions  v^,  vi\  .  .  .Vi^'"'^  belong  to  different  lines  of  the 
table  Tq).  It  appears  therefore  that  the  p.  lines  of  To)  coincide  in 
sets  of  k  each.     We  have  then 

Theorem  XIT.  Every*  individual  cycle  of  order  k  which 
occurs  in  (2),  and  consequently  every  one  which  occurs  in  H,  gives 
rise  by  transformation  with  respect  to  all  the  h  substitutions  of  H, 

to  ~  cycles.     The  h'  distinct  conjugate  cycles  of  order  k  which  occur 
k 

in  H  therefore  give  rise  to  -t-  =  —  cycles.     The  number  of  letters  oc- 

fc        fc 

curring  in  these  cycles  is  therefore  equal  to  h,  the  order  of  H.  From 
this  it  follows  that  the  number  of  letters  in  all  the  cycles  of  order  k 
is  a  multiple  of  the  order  of  H.  The  multiplier  is  the  number  of 
sets  of  non-conjugate  cycles  of  order  k. 


MULTIPLE-VALUED    FUNCTIONS ALGEBRAIC    RELATIONS.  57 

Corollary.  The  number  of  elements  which  remain  un- 
changed in  the  several  substitutions  of  H  is  a  multiple  of  the  order 
of  H.  If  every  element  can  be  replaced  by  every  other  one  by  the 
substitutions  of  H,  this  number  is  exactly  equal  to  the  order  of  iJ.* 

Example.     Consider  the  alternating  groups  of  four  elements, 

\^l)  {^2}  {^3/  ('^4))       (^1^2)  (^3^4)7       ('^1^3/  {^■2'^i)j       ('^1^4)  (^2^3)> 
[Xi)  (a?2^3^4)?  '•^2)   (^1^3^4)5        (^3)   (^1^2^4)5       (^4)  V^1^2^3)j 

[Xi)  [X^X^X^),  (^2)  \'^l'^4'^3)j       ('^^3)  C^l^4'^2))      V^'i)  (^i^3'^2J' 

Here  the  number  of  cycles  with  one  element  is  12,  which  is 
equal  to  the  order  of  the  group.  The  number  of  elements  which 
occur  in  cycles  of  the  second  order  is  also  12.  But,  for  A;  =  3,  the 
number  of  elements  is  24  =  2  •  12.  Correspondingly  it  is  readily 
shown  that  the  group  permits  of  replacing  any  element  by  any 
other  one;  that  the  cycles  of  order  2  are  all  conjugate;  but  that 
the  cycles  of  order  3  divide  into  two  sets  of  four  each: 

\«^i«^2'^3/j     yp^i^'i'^i/i     \XiX^X2)j     \X2X^Xr^^f 

\"^l"^3'^2/5         V'^1'^4'^3J)         \'^l«^2'^4/?         \'^2'^3'^4J) 

the  second  set  being  non- conjugate  with  the  first. 

§  52.  We  return  now  to  the  table  constructed  in  §  45.  This 
table  did  not  possess  the  last  of  the  four  properties  noted  in  §  41 ;  the 
substitutions  of  one  line  were  not  necessarily  all  different  from  those 
of  the  other  lines.  For  every  group  certainly  contains  the  identical 
substitution  1,  which  therefore  occurs  p  times;  and  again  in  the 
example  of  §  46  three  other  substitutions 

'  'A  . 
{^XiX2)  [X^X^Jj         {X1X2)  (^£<y2'^4/)         V'^1'^4/  \'^2'^3) 

occur  in  each  of  the  three  groups.  We  have  now  to  determine  in 
general  when  it  is  possible  that  one  and  the  same  substitution  shall 
occur  in  all  the  groups  Gi,G2,  .  .  .  Gp  belonging  respectively  to  the 
several  values  <Pi,<P2,  -  -  -  <Pp  of  <p.  We  shall  find  that  the  example 
just  cited  is  a  remarkable  exception,  in  that  there  is  in  general  no 
substitution  except  1  which  leaves  all  the  values  of  a  function  un- 
changed, f 

♦In  connection  with  this  Section  C/.  Frobenius,  Crelle  CI.  p.  273,  followed  by  an 
article  by  the  Author,  ihid.  CIII  p.  32L 

t  L.  Kronecker:    Monatsberichte  d.  Berl.  Akad.  1879,  208. 


58  '  THEORY    OF    SUBSTITUTIONS. 

If  we  apply  to  the  series  of  functions  95] ,  ^2  >  -  •  -  ^p  any  arbitrary 
substitution  t,  we  obtain 

These  values  must  coincide,  apart  from  the  order  of  succession,  with 
the  former  set,  for  ^j ,  ^^2?  -  -  •  ^p  are  all  the  possible  values  of  (f,  and 
the  p  values  just  obtained  are  all  different  from  one  another.  The 
groups  which  belong  to  the  latter 

(j~^G^<T^   (j^^G^G^  a~^G^(T,  .  .  .  <T~^Gp(y 

are  therefore  also,  apart  from  the  order  of  succession,  identical  with 
Gi,  Go,  G^,  .  .  .  Gp,  that  is,  the  system  of  the  (r's,  regarded  as  a  whole, 
is  unaltered  by  transformation  with  respect  to  any  substitution 
whatever.  If  now  we  denote  by  H  the  group  composed  of  those 
substitutions  which  are  common  to  G^^G^,  ...  Gp,  then  H  is  also  the 
group  of  those  substitutions  which  are  common  to  <t~^Gi(t,  ff~^6r2<J", 
.  .  .  <T~'^Gp(T.  But  the  latter  group  is  also  of  course  expressed  by 
(T~^H(T-,  consequently  we  have 

that  is,  the  group  H  is  unaltered  by  transformation  with  respect  to 
any  substitution;  it  includes  therefore  all  the  substitutions  which 
are  similar  to  any  one  contained  in  it. 

We  proceed  now  to  examine  the  nature  of  a  group  H  of  this 
character.  We  consider  in  particular  those  substitutions  of  H 
which  affect  the  least  number  of  elements,  the  identical  substitution 
excepted.  It  is  clear  that  these  can  contain  only  cycles  of  the  same 
number  of  elements,  since  otherwise  some  of  their  powers  would 
contain  fewer  elements,  without  being  identically  1. 

We  prove  with  regard  to  these  substitutions,  first  that  no  one  of 
their  cycles  can  contain  more  than  three  elements.  For  if  H  con- 
tains, for  example,  the  substitution 

and  if  we  take  (t  =  (x^x^),  then,  since  (t~^H(t  =  H,  the  substitution 

will  also  occur  in  H.  Now  s^  only  differs  from  s  in  the  order  of  the 
two  elements  x^  and  a?^.  Consequently  their  product,  which  must 
also  occur  in  H,  since  If  is  a  group, 


MULTIPLE- VALUED    FUNCTIONS ALGEBEAIC    RELATIONS.  59 

o  •  Sj  —  ( eA/3  )   yOC-^OC^  «  .  .  )   .  •  • 

will  certainly  not  afPect  the  element  x^,  but  cannot  be  the  identical 
substitution,  because  it  contains  at  least  the  cycle  (x^Xi . .  . ).  This 
product  therefore  affects  fewer  elements  than  s^  which  is  contrary  to 
hypothesis. 

Secondly  we  prove  that,  if  n  >  4,  the  substitution  of  H  which 
affects  the  least  number  of  elements  cannot  contain  more  than  one 
cycle.     For  otherwise  H  would  contain  substitutions  of  the  form 

and  therefore  the  corresponding  conjugate  substitutions  with  respect 

to    'T  =   (x^iCg) 

Sfi   =■  {XiX2)  yX^X^)  .  .  .  ,       Sp   =^  (^XiX.2X^)  [X^X^Xq)  .  .  . 

Consequently  the  corresponding  products 

Sa''Sa'=  (X,)  {X2)  (a?3  ...)...  ,     Sp-'Sp'  =  (X,)  (^2)  (0:^3)  (x^Xr^X^)  .  .  . , 

which  are  not  1,  but  affect  fewer  elements  than  s,  must  also  occur  in 
H,  which  would  again  be  contrary  to  hypothesis. 

If  then  n  >  4,  either  H  consists  of  the  identical  substitution  1, 
or  H  contains  a  substitution  (Xf^x^),  or  a  substitution  {x^x^Xy).  In 
the  second  case  H  must  contain  all '  the  transpositions,  that  is  H  is 
the  symmetric  group.  In  the  third  case  H  must  contain  all  the  cir- 
cular substitutions  of  the  third  order,  that  is  H  is  the  alternating 
group.     (Of.  §§  34-35). 

Returning  from  the  group  H  to  the  group  G,  it  appears  that  if 
Gi,  0-2,  .  .  .  Gp  have  any  substitution,  except  1,  common  to  all,  then 
either  the  second  or  the  third  case  occurs.  H,  which  is  contained  in 
G,  includes  in  either  case  the  alternating  group;  G  is  therefore 
either  the  alternating  or  the  symmetric  group,  and  p  =  2,  or  p  —  1. 

If,  however,  n  =  4  we  might  have,  beside  s^  =  1,  another  substi- 
tution 

in  the  group.     With  this  its  conjugates,  of  which  there  are  only  two, 

§2  :=  [XiX'j)  {X2Xi),       Si  =  {XiX^)  {X2X^)j 

must  also  occur.  The  group  H  cannot  contain  any  further  substitu- 
tion without  becoming  either  the  alternating  or  the  symmetric  group. 
.We  have  then  the  exceptional  group 


A 


60  THEORY    OF    SUBSTITUTIONS. 

and  this  actually  does  transform  into  itself  with  respect  to  every  sub- 
stitution. Returning  to  the  group  G  it  follows  from  §  43,  Theorem 
II,  that  the  order  of  (r  is  a  multiple  of  that  of  H,  that  is,  a  multiple 
of  4;  again  from  Theorem  II  the  order  of  G  must  be  a  divisor  of 
u  4!  =  24.     The  choice  is  therefore  restricted  to  the  numbers  4,  8,  12, 

and  24.  The  last  two  numbers  lead  to  the  general  case  already  dis- 
cussed where  p  =  2,  or  1.  The  first  gives  G  =  H,  />  =  6,  and  for 
example, 

(p2  —  \X-iX^  -\-  Xt^X^J  \XiX^  -\-  i3?2«^3/?     ^4  "~"  V'^1'^3     \     '^2'^^)  [XiXo,  ~\~  »^3'^4/ 

^5  ~~  \XiX^  -\-  X-iX-^J  \X1X2  ~\~  '^3'^4 j?     ^6  ~~  \X]X^  -\~  X2X^j  yXiX^  "T"  «^2'^4/  • 

In  the  second  case,  ?^  =  8,  G  contains  H  as  a  subgroup.  To  obtain 
G  we  must  add  other  substitutions  to  those  of  H.  None  of  these 
can  be  cyclical  of  the  third  order,  for  in  this  case  we  should  have 
r  =  12  or  24.  If  we  select  any  other  substitution,  we  obtain  the 
group  of  §  46,  which  is  included  among  those  treated  in  §  39.     For 

this  group  p  =  3,  and,  for  example, 

f 

/ft      /y»    /'v*         I       ry*   rv*  /ft      /v»    rv*         i       ry*   /y*  /ft      ^V»    rg*         I        y*   /y* 

V   1    tA/-^tA^2       1^   tAy^tAy^  ,  V  2   tt/JeA/3        \^  tA/O'^i  J  V  3    tAyji4y^       |^   t4y2«^3  • 

Theorem  XI.     If  n  ^4:  there  is  no  function,  except  the  al- 
ternating and  symmetric  functions,  of  ivhich  all  the  p  values   are 
unchanged  by  the  same  substitution  (excluding  the  case  of  the  identi- 
cal substitution).    7fw  =  4,  all  the  values  of  any  function  belong 
ing  to  the  same  group  with 

<p  =  {x^X2  +  ^3^4)  —  (^1^3  +  XiX^  or  ivith  4'  =  ^1^2  +  ^3^4 
are  unchanged  by  the  substitutions  of  the  group  i^ 

H  =  [1,   (X^X2)  (X^X,),   (X^X^)  (x2Xi),  (XiXi)  {X2X^)]. 


§  53.  We  have  thus  far  examined  the  connection  between  the 
values  of  a  />- valued  function  from  the  point  of  view  of  the  theory  of 
substitutions.  We  turn  now  to  the  consideration  of  the  algebraic 
relations  of  these  values. 

We  saw  at  the  beginning  of  the  preceding  Section  that  the  sys- 
tem of  values  ^1,  ^'2?  •  •  •  ^p  belonging  to  a  function  <p  is  unchanged 


MULTIPLE- VALUED    FUNCTIONS ALGEBRAIC    RELATIONS.  61 

as  a  whole  by  the  application  of  any  substitution,  only  the  order  of 
succession  of  the  several  values  being  altered.  All  integral  symmet- 
ric functions  of  f  i ,  c^9 ,  .  .  .  ^n  are  therefore  unchanged  by  any  substi- 
tution, and  are  consequently  symmetric  not  only  in  the  ^'s  but  also 
in  the  Xx^.  They  can  therefore  be  expressed  as  rational  integral 
functions  of  the  elementary  symmetric  functions  Ca  of  the  o^^'s.  If 
we  write  then 

S(cri)  =fi  +  9^2+  .  .  .  +^P    =    i?i(Ci,C2,.  ..C„), 

S  (cTj  9-2)  =  9^1  CTo  -f  C^jCg  H-    .   .   .  =     i^2  (Ci  ,  C2  ,   .   .   .  C„), 

S(cfiS^2  .  .  .  f>)  =  rifefa  .  .  •  S^p  =   ^-^pC^i,  C.,  .  .  .C,), 

the  -R's  are  the  coefficients  of  an  algebraic  equation  of  which 
S^i  5  S^2  ?  •  •  •  ^P  a^^6  the  roots. 

Theorem  XII.     The  />  values  c^i,  t^^,  .  .  .  (pp   of    a   p-valued 
integral  rational  function  (f  are  the  roots  of  a7i  equation  of  degree  p 

the  coefficients  of  ivhich  are  rational  integral  functions  of  the  ele- 
mentary symmetric  functions  Cj ,  Co,  . . .  c„  of  the  elements  Xi,X2,  ...  x,,. 

§  54-.     As  an  example  we  determine  the  equation  of  which  the 
three  roots  are 

O-j^  —  Ju^JLo     I     t^gt-C'^  ^        y- 2  —        13     I     «^2*^4  5  *  3  —  Ju-^JL^     |     tX'2*^3  5 

where  Xi,X2,  x^,  x^  are  themselves  the  roots  of  the  equation 
fix)  =  a?^ — CjX*^  -|--  <^2'^^  -  ^3^^  +  ^4  =  0. 
We  find  at  once 

fx  -V  f  2  +  f^3  =   S  {X^X^  =   C2; 

and  again,  by  §  10,  Chapter  I, 

^iS^2  +  9'2^3  +  f  3f  1  =  S  (a^i^^a^a)  =  ^^^4  +  ^-CiCg  +  ^-Cgl 

The  numerical  coefficients  a,  /?,  y  are  readily  found  from  special  ex- 
amples.    They  are  «  =  — 4,  ,5  =  1,  ^  =  0.     Hence 

^1^2  +  ^2?'3  +  f%{\  —  C1C3  —  4  C4  . 

Finally 

^'X^l'fz  —  ^Vm?!  X2  X^  I  ~j~  XiX2X^X^  OliCj   ) 
"~~  ^1  ^4  4C2C4  -J-  C3  , 

Accordingly  the  required  equation  is 


62  THEORY    OF    SUBSTITUTIONS. 

/(^)  =  ^'  — C2^'  +  (C1C3  — 4c,)^  —  (ci'c^  — 4c,C4  +  Cg'O  =  0.     ^ 

We  examine  the  discriminant  of  this  equation,  i.  e.,  of  its  three 
roots.  To  determine  this  function  it  is  not  necessary  to  employ  the 
the  general  formula  obtained  in  §  10,  Chapter  I.     We  have  at  once 

<Pl  —  <P2=  (^1  —  ^4)  (^2  —  ^^3), 
<f2  —  <P3=  (^1  —  ^2)  (^3  —  ^i), 

<P^  —  <Pi  =  (a?i — X.,)  (a?2  —  X,), 
and  consequently,  if  we  denote   the  discriminant  of  the  f^'s  by  J^ 
and  that  of  the  x^s  by  J, 

^9=  (<Pi—<f2y{<P2—ny(<f3—<PiY  - 

—  \"^  1        X2)   \Xi        «^3/   \«^i        X^j   \X2        X^f   \X2        x^)   {x^        x^j    ^^^  iJ. 

We  observe  here  therefore  that  the  discriminant  of  an  equation 
of  the  fourth  degree  can  also  be  represented  in  the  form  of  the  dis- 
criminant of  an  equation  of  the  third  degree.  A  more  important 
consideration  is  that  the  special  theorem  here  obtained  can  be  gen- 
eralized in  another  direction,  to  which  we  next  turn  our  attention. 

§  55  We  start  out  from  the  table  of  §  41.  If  (p  is  not  single- 
valued,  the  first  line  of  this  table,  i.  e.,  the  group  G^  belonging  to  <Pi 
does  not  contain  all  the  transpositions.  If  a  transposition  (XaX^) 
occurs  in  the  second  line,  it  results  from  the  construction  of  the 
table  that  (XaX^)  converts  ^1  into  ^2-  If  therefore  Xa=  x^,  then 
fi  =  <P2  also,  and  consequently  <p^  —  (p^,  since  it  vanishes  for  x„  =  x^, 
is  divisible  by  Xa — x^. 

If,  then,  any  transposition  (XaX^)  does  not  occur  in  the  group  G 
of  (fi ,  one  of  the  differences  <Pi — cta  (^^  =  2, 3,  .  .  .  f))  is  divisible  by 
a  factor  of  the  form  Xa  —  Xp. 

')i(}i, 1 ) 

Now  there  are  in  all  — ^ — -  transpositions  of  71  elements.   If  the 

first  line  of  the  table,  i.  e.,  G^  contains  exactly  qof  these,  then  the 
other  lines  contain  ,-^ q.    The  product  (fi  —  s^2)(^i  —  9^3)  ••  • 

Yl(  Yi 1 ) 

(9^1  —  ^p)  is  therefore  divisible  by ^ — - — q  different    factors    of 

the  form  Xa — x^,  and  therefore  by  their  product. 

Instead  of  starting  out  with  the  value  f^i,  we  might  equally  well 
have  taken  9^2-     Since  the  group  G^  —  <^2~^G^\'^2  belonging  to  9^2  =  9V. 


MULTIPLE-VALUED    FUNCTIONS ALGEBKAIC    KELATIONS.  ^63 

is  similar  to  the  group  Gi ,  it  also  contains  q  transpositions,  and  the 
product  {ip^  —  (pi)  (^2  —  ^3)  .  •  .  (^2  —  ^p)  is  also  divisible  by —^-^ q 

factors  of  the  form  Xa — x^.  The  same  reasoning  holds  if  we  start 
with  ^3 ,  ^4 .  .  .  s^p .  If  we  multiply  the  separate  products  together, 
we  find  that 

^'•.{<P\  —  <Pp)\ 

[yiyYl \_\  ~\ 

rt..;^  - — g    factors  a?a — x^.      But 

since  J^  is  symmetric  in  the  a^x's,  the  presence  of  a  factor  cCa — x^ 

requires  that  of   every  other  factor  Xy  —  a?5,  and  consequently  of 

A=  II  {i)Ca — x^\  the   discriminant  of  f(x).      Suppose  that  A^  is 

a>ft 

the  highest  power  of  J  which  is  contained  as  a  factor  in  J^,  then,  as 
A  contains  n{n  —  1)  factors  Xa  —  x^,  and  consequently  zl'  contains 
n{n  —  1)^  such  factors,  we  must  have 


„(„_,),>,[^fcl)-g], 


t  >  ''  '^'' 


2       n(re  — 1) 

nin  - 1^ 

The  number  t  can  be  0  only  when  q  =  ,  that  is,  when  all 

the  transpositions  occur  in.  G^.  c?  is  then  symmetric  and />  =  1. 
Again  q  can  be  0  only  when  G  contains  no  transposition.  One  of 
the  cases  in  which  this  occurs  is  that  where  G  is  the  alternating 
group  or  one  of  its  subgroups. 

Theorem  XII.  //  <p  is  p -valued  function  of  the  n  elements 
Xi,X2,  .  .  .  .T„ ,  the  group  of  which  contains  q  transpositions,  the  dis- 
criminant A^  of  the  p  values  of  f  is  divisible  by 

If  (p  is  not  symmetric,  the  exponent  of  A  is  not  zero.  If  the  group 
of  if  is  contained  in  the  alternating  group,  q  is  zero. 

All  multiple-valued  functions  therefore  have  some  of  their  vahies 
coincident  if  two  of  the  elements  X\  become  equal. 


64  THEORY    OF    SUBSTITUTIONS. 

We  perceive  now  why  it  was  impossible  (§  32)  to  obtain  n!-valued 

functions  when  any  of  the  elements  .r^  were  equal  {Cf.  also  §  104). 

§  56.     Returning  now  to  the  equation  (§  53)  of  which  the  roots 

are  <p^,(f^,  •  •  •  ^p, 

we  endeavor  to  determine  whether  and  under  what  circumstances 
this  equation  can  become  binomial,  i.  e.,  whether  there  are  any 
/>-valued  functions  whose  />*^^  powers  are  symmetric.  For  (>  =  2,  we 
already  know  that  (p=  \f  d  satisfies  this  condition.  In  treating  the 
general  case  we  will  assume  that,  if  the  required  function  (p  contains 
any  factors  of  the  form  V  J ,  these  factors  are  all  removed  at  the 
outset.     If  the  resulting  quotient  is  ^'',  so  that 

then,  since  cp-p  and  (V-j)"^  ai'^  both  symmetric,  v''"^  is  symmetric 
also.     We  write  then 

If  d\  be  any  root  of  this  equation,  and  if  m  be  a  primitive  (2/>)^^  root 
of  unity,  then  all  the  roots  are 

and  consequently 

From  Theorem  XIII  this  discriminant  must  be  divisible  by  J,  unless 
4>  is  itself  symmetric.  But  the  factors  containing  w  are  constant 
and  therefore  not  divisible  by  J,  and  by  supposition  (l\  does  not  con- 
tain V  -^  as  a  factor.  Consequently  v''i  is  symmetric,  and  we  have, 
according  as  a  is  odd  or  even, 

<P  =  S,     ^  =  S\/J. 

Theorem  XIV.  If  the  n  elements  x^^Xi,  .  .  .  x„are  inde- 
pendent, the  only  unsymmetric  functions  of  ichich  a  power  can  be 
symmetric  are  the  alternating  functions. 

§  57.  On  account  of  the  importance  of  this  last  proposition  we 
add  in  the  present  and  following  Sections  other  proofs  based  on 
entirely  different  grounds. 


MULTIPLE-VALUED    FUNCTIONS ALGEBRAIC    RELATIONS.  65 

If  w  is  a  primitive  p^^  root  of  unity,  and  if  (f-^  is  any  root  of  tlie 
equation 

<f9  —  S  =  0, 

then  all  the  roots  of  this  equation  are 

Since  the  w's  are  constants,  all  the  values  of  <p  have  the  same  group. 
From  Theorem  XI,  this  must  be,  for  ^  ^  4,  either  the  symmetric  or 
the  alternating  group,  or  the  identical  operation  1.  The  first  two 
cases  give  />  =  1  or  2.  In  the  last  case  p  =  n\.  All  the  values  of  a 
function  are  of  the  same  type,  and  consequently  there  are  substitu- 
tions which  transform  one  into  another.  Suppose,  in  the  case 
p  =  n\,  that  T  converts  the  value  <Pi  into  ^2  =  "^^i;  then  (t^  converts 
<Pi  into  oj^  <pi .    Accordingly  the  series 

consists  of  distinct  substitutions,  which  therefore  include  all  the  nl 
possible  substitutions,  (t  must  therefore  be  a  substitution  of  degree 
n  and  order  n ! .     Such  a  substitution  does  not  exist  if  ti  >  2. 

The  case  n  =  4  furnishes  no  exception.  In  this  case  the  group 
common  to  all  the  values  of  ^  might  be  the  special  group  (Theorem 
XIII) 

Ct  =  [_i,  (a?i;r2J  (CC3X4),    {XiX^)  (£^2^4))    (^1^4)  (^2^3)J  j 

<p  would  then  be  a  six-valued  function,  and  there  must  be  a  substi- 
tution <7  which  converts  <pi  into  oxp^  and  which  is  of  order  6.  But 
there  is  no  such  substitution  in  the  case  of  four  elements. 

§  58.  Finally  we  give  a  proof  which  is  based  on  the  most  ele- 
mentary considerations  and  which  moreover  leads  to  an  important 
extension  of  the  theorem  under  discussion. 

In  the  first  place  we  may  limit  ourselves  to  the  case  where  />  is  a 
prime  number.  For,  if  p=p.q,  where  _p  is  a  prime  number,  it 
follows  from 

that  there  is  also  a  function  (p^  of  which  a  prime  power,  the  p^^ ,  is 
symmetric. 

If,  accordingly,  we  denote  by  ^  a  function  of  which  a  prime 
power,  the  p^^,  is  symmetric   while  <p  itself  is  unsymmetric,   then, 


66  THEORY    OF    SUBSTITUTIONS. 

since  the  group  of  ^  cannot  contain  all  the  transpositions  (§  34), 
suppose  that  (T  =  (XaXp)  converts  ^^  into  <f^,  where  <p^==^^.     Since 

it  follows  that 

where  <y  is  a  primitive  p^^  root  of  unity.  If  we  apply  the  substitu- 
tion (T  again  to  the  last  equation,  remembering  that  (t-  =  1  and  that 
consequently  <p^'-  =  <Pi ,  we  obtain  the  new  equation 

Multiplying  these  two  equations  together  and  dividing  by  <Pi  <fa- ,  we 
have 

<.'-■= 1, 

and  consequently,  since  p  is  a  prime  number,  p  =  2  and  <p  =  S \/  J- 
Having  shown  that  only  the  alternating  functions  have  the  prop- 
erty that  their  prime  powers  can  be  symmetric,  we  may  next  exam- 
ine whether  there  are  any  functions  prime  powers  of  which  can  be 
two-valued. 

Suppose  that  (/>  is  multiple-valued,  while  its  q^^  power  is  two-val- 
ued, q  being  prime.  Then  there  is  some  circular  substitution  of 
the  third  order  (t  =  (xaX^Xy),  which  does  not  occur  in  the  group  of  (I\ 
since,  if  this  group  contains  all  the  substitutions  of  this  form,  it 
must  be  the  alternating  group  (§  35).  Suppose,  then,  that  (,'v4"V''iv 
but  that 

since  v''",  being  a  two-valued  function,  is  unaltered  by  a  circular  sub- 
stitution of  the  third  order.     We  must  therefore  have 

where  w  is  not  1  and  must  therefore  be  a  primitive  q^^  root  of  unity. 
If  we  apply  to  this  last  equation  the  substitutions  <t  and  <7-,  and 
remember  that  ^^  =  1,  and  that  consequently  v'v^*  =  v''i  >  we  obtain 

Multiplying  the^e  three  equations  together  and  removing  the  func- 
tional values,  wG  have 

a.«=rl,  g  =  3. 


MULTIPLE-VALUED    FUNCTIONS ALGEBRAIC    RELATIONS.  67 

If  now  WG  assume  n  >  4,  then  the  group  of  (J'  cannot  contain  all 
the  circular  substitutions  of  the  fifth  order,  (Theorem  X,  Chapter 
II).     If  r  is  one  of  those  not  occurring  in  the  group  of  (J',  then 

V''T+V''n  but 

and  consequently,  if  ^^  be  a  q^^  root  of  unity  different  from  1, 


It  follows  from  this,  precisely  as  above,  that,  since  r"  =  1, 

and  consequently  o>^  =  1,  g  =  5. 

But  this  is  inconsistent  with  the  first  result.  It  follows  there- 
fore that  n  is  not  greater  than  4. 

Theorem  XVII.  If  n>  A^  there  is  no  multiple-valued 
function  a  power  of  ivhich  is  two-valued,  if  the  elements  x  are 
independent  quantities. 

§  59.  We  conclude  these  investigations  by  examining  for  n^4 
the  possibility  of  the  existence  of  functions  having  the  property  dis- 
cussed above. 

The  case  n  =  2  requires  no  consideration.  In  the  case  n  =  3  we 
undertake  a  systematic  determination  of  the  possible  functions  of 
the  required  kind.     We  begin  with  the  type 

and  attempt  to  determine  «,  /5,  y  so  as  to  satisfy  the  required  condi- 
tions. For  this  purpose  we  make  use  of  the  circumstance  that  some 
<T  =  {x^x.2X.^)  converts  (p^  into  oj<fi  (w^  =  1)  so  that 

<p^  —  aXi  -f  fe'"  +  yx^''  =  a){ax{  -f  ^X^'  +  yXo^'), 
Y  =  wa ,  /5^  ujy  =  oj^a,  a  =  <y/5  zzz  lu'^y  =:  w'^a  =.  a. 

The  last  three  equations  can  be  consistently  satisfied  for  every  value 
of  a.     We  may  take  «  —  1 ;  and  therefore 

<Pi  =  a?i''  -f  -  0)^X2'  +  ^^s' 
is  a  function  of  the  required  type. 

This  result  is  confirmed  by  actual  calculation.     We  find 


68  THEORY    OF    SUBSTITUTIONS. 

(a?i'''  +  xi'-  +  xt)  +  6 x{x{x{—  Y {^I'^i'  +  ^i"3c{ -\- xi''x{  +  . . . ) 
3       

Z 

If  r  =  1  the  result  becomes  simpler,  in  that  the  last  parenthesis 
becomes  equal  to 

^J  =  (i(?j        X2)  \Xc2^        Xo^)\X'^        X3); 

whereas,  in  general,  this  parenthesis  is  only  a  rational  function  of 
\/ A.     If  we  write,  as  usual, 

X\  ~\~  X-2  "I    X<^  —  Cj ,     XyJC^    1    X2X^  ~j~  X^X-^  —  C2 ,     X-yXoX'^  —  C3  , 

we  have,  for  r  =  1, 


c^  =  J  j  2c/ — ^c,C2  +  27c3±3V— 3JS. 
Suppose  now  that  n  =  4. 

It  is  obvious  that  a  function  of  the  type  ax{  -\-  [iXo  +  T^{  +  ^^l\ 

in  which  every  term  contains  only  one  element,  cannot  satisfy  the 

condition  of  being  multiplied  by  w  when  operated  on  by  <r  =  {x^x^x^. 

We  enquire  then  whether  the  required  function  can  be  of  the 

type 

Ci  =  ax{x.{  4-  ?^Xi'x{  +  r^{x{  +  Xl'{a^x{  +  ^^X{  +  n^s''), 

where  every  term  contains  two  of  the  four  elements.  If  this  type 
should  also  fail  to  satisfy  the  requirements  we  should  have  to  pro- 
ceed to  more  complicated  forms.  We  shall,  however,  obtain  a  posi- 
tive result  in  the  present  case,  and  in  fact  we  may  take  r  =  1,  so  that 

(f„  —  ax^x^  +  fe^i  +  yXiX^  +  a^4(«i^2  +  /?i^3  +  YiXiY 
From  the  condition  c?<y=w9?i,  we  have  the  series  of  equations 

All  of  these  equations  are  satisfied  independently  of  the  values  of  a 
and  «! ,  and  we  have 

^j  =z  a[XiX2  +  6y^a?2^3  +  loX^Xi)  -{-  a^^XiX^  -j"  w'il?2^4  +  tt^X^Xi). 

But  again,  the  substitution  r  =  (a;ia;2a74)  converts  ^j  into  ^t>  where 
y^T  is  equal  to  the  product  of  <pi  by  some  cube  root  of  unity,  since 


MULTIPLE-VALUED    FUNCTIONS ALGEBEAIC    RELATIONS.  69 

9\  —  ^T'  Whether  this  cube  root  is  1,  w,  or  co^  cannot  be  deter- 
mined beforehand.     We  find 

and  since  the  terms  of  ^i  and  ^^  which  contain  x^x^  are 

a^x^x^^  and  a^ixP'x^x^^ 

the  cube  root  of  unity  by  which  (p^  is  multiplied  must  therefore  be 
o?.  From  the  equation  ^^  =  ^"f*!  it  follows,  by  comparison  of  coef- 
ficients that 

These  two  equations  are  consistent,  since  fy^  =  1.  Putting  «  =  1,  and 
arranging  according  to  the  powers  of  w,  we  have  then 

<Pi  =  {x^X2  +  x^^x^  +  w(a?ia?3  +  x^x^)  +  u)\x^x^  +  x^x^. 

The  function  <p^  is  therefore  a  combination  of  the  three  values  of 
a  function  which  we  have  already  discussed.     The  group  of  (f^  is 

Cr  :=  |_lj    \X1X2)  (^3^4))    \XiX-i)  (CC2CC4),    {XiX^)  (a?2^3)J. 

That  «>/  is  two- valued  is  also  readily  shown,  if  we  write 

X]X2  ~\~  oc^x^  —  2/1 )  XiXq  -J-  x.2X^  ^=:  2/2 )  x^x^  -J-  x^x^  ^  2/3  • 

For  then  (p^  coincides  with  the  expression  obtained  above  for  the. 
case  91  =  3 ;  and  since  2/1  ?  2/2  5  2/3  >  ^^^  the  roots  of  the  equation 

y' — C2  2/'  +  (C1C3 — 4cJ  2/ — {c;\ — 4C2C4  +  C3'')  =  0, 

where  the  c's  are  the  coefficients  of  the  equation  of  which  the  roots 
a?i,  a?2,  a?3,  a?4,  (§  54),  we  can  translate  the  expression  obtained  for 
w  =  3  directly  into    a  two- valued   function   of    the  four   elements 

a?i,  a?2,  a?3,  CC4,  since  we  have  (§  54)  A,,  =  J^. 


CHAPTER  IV. 


TKANSITIVITY   AND  PRIMITIVITY.      SIMPLE  AND  COMPOUND 
GROUPS.    ISOMORPHISM. 

§  60.     The  two  familiar  functions 

ry^   /y»     L.  /y*   ^y*  -'V   O^    O^   O^ 

differ  from  each  other  in  the  important  particular  that  the  group 
belonging  to  the  former 

G  =  [1,  (XYJC^),  (x^Xi),  (X^X,)  (x^Xi),  (X^X^)  {X,X,),  (XiX,)  (x^x-^), 
{XiX^X2X^)j  {XlX^X2XQ)^ 

contains  substitutions  which  replace  x^  by  X2 ,  Xr^ ,  or  x^,  while  in  the 
group  belonging  to  the  latter 

G,  =  [1,  (x,X2),  {x,Xi),  (x.x^)  (x,x^)'] 

there  is  no  substitution  present  which  replaces  x^  by  x^  or  x^.  The 
general  principle  of  which  this  is  a  particular  instance  is  the  basis 
of  an  important  classification.  We  designate  a  group  as  transitive, 
if  its  substitutions  permit  us  to  replace  any  selected  element  x^  by 
every  other  element.  Otherwise  the  group  is  intransitive.  Thus  6r, 
above,  is  transitive,  while  Gi  is  intransitive. 

It  follows  directly  from  this  definition  that  the  substitutions  of 
a  transitive  group  permit  of  replacing  every  element  Xi  by  every 
element  x,,.  For  a  transitive  group  contains  some  substitution 
s  =  (xiXi ...)...  which  replaces  x^  by  Xi,  and  also  some  substitu- 
tion t=  {x^Xk  ...)...  which  replaces  x^  by  x,,.  Consequently  the 
product  s~^t,  which  also  occurs  in  the  group,  replaces  x,-  by  a?^. 

The  same  designations,  transitive  and  intransitive,  are  applied  to 
functions  as  to  their  corresponding  groups. 

It  appears  at  once  that  the  elements  of  an  intransitive  group 
are  distributed  in  systems  of  transitively  connected  elements.  For 
example,  in  the  group  G^  above  a?,  and  0^35  and  again,  x^  and  0^4,  are 
transitively  connected.  Suppose  that  in  a  given  intransitive  group 
there  are  contained  substitutions  which  connect  x^  X2,  .  , .  Xa  transi- 


GENERAL    CLASSIFICATION    OF    GROUPS.  il 

tively,  others  which  connect  x^+i,  x„^2i  •  •  -  ^a  +  bj  and  so  on,  but 
none  which,  for  instance,  replace  x^  hj  x„^\  (^  =  1)5  and  so  on.  The 
maximum  possible  number  of  substitutions  within  the  several  sys- 
tems is  a ! ,  6 ! ,  .  .  . ,  and  consequently  the  maximum  number  in  the 
given  group,  if  a,  b,  .  .  .  are  known,  is  a!  5!  .  .  .  If  only  the  sum 
a-\-b-\-  ...  =n  is  known,  the  maximum  number  of  substitutions  in 
an  intransitive  group  of  degree  n  is  determined  by  the  following 
equations : 

{n—l)l  1!  =  ^^^(n— 2)!  2!  >  {n  —  2)l  2!,     {n  >  3) 

f, 9 

{n-2)l2l=^^{n  —  ^)\^\>{n  —  ^)\^\,     (n>5). 

Theorem  I.  The  maximum  orders  of  intransitive  groups 
of  degree  n  are 

(n—1)!,  4(^—1)!,  {n—2y.  2l,{n  —  2)\,  (n  — 3)!3!,  (w  — 3).!2!,  .  .  . 

The  first  two  orders  here  given  correspond  to  the  symmetric  and 
the  alternating  groups  of  (n — 1)  elements,  so  that  in  these  cases  one 
element  is  unaffected.  The  third  corresponds  to  the  combination 
of  the  symmetric  group  of  (n  —  2)  elements  with  that  of  the  re- 
maining two  elements.  The  fourth  belongs  either  to  the  combina- 
tion of  the  alternating  group  of  (n — 2)  with  the  symmetric  group 
of  the  remaining  two,  or  to  the  symmetric  group  of  {n  —  2)  ele- 
ments alone,  the  other  two  elements  remaining  unchanged;  and  so 
on. 

The  construction  of  intransitive  from  transitive  groups  will  be 
treated  later,  (§  99). 

§  61.  We  proceed  now  to  arrange  the  substitutions  of  a  transi- 
tive group  in  a  table.  The  first  line  of  the  table  is  to  contain  all 
those  substitutions  7 

Sj  =   1 ,    §2 ,    S3  ,   .   .   .  S„j 

which  leave  the  element  x^  unchanged,  each  substitution  occurring 
only  once.  From  the  definition  of  transitivity,  there  is  in  the  given 
group  a  substitution  t.,  which  replaces  x^  by  'X2.  For  the  second  line 
of  the  table  we  take 


72  THEORY    OF    SUBSTITUTIONS. 

We  show  then,  1)  that  all  the  substitutions  of  this  line  replace  x^  by 
X2;  for  every  s^  leaves  x^  unchanged  and  (t^  converts  a?i  into  X2',  2) 
that  all  the  substitutions  which  produce  this  effect  are  contained  in 
this  line;  for  if  r  replaces  x^  by  x.2,  "^^2"^  leaves  x^  unchanged  and  is 
therefore  contained  among  the  s^'s;  but  from  t(T2~^  —  s^  it  follows 
that  r  =  s^^.2 ;  3)  that  all  the  substitutions  of  the  line  are  distinct ; 
for  if  Saf^2  =  s^  ''■2  ?  we  obtain  by  right  hand  multiplication  by  't,"  \ 
Sa  =  Sp',  4)  that  the  substitutions  of  the  second  line  are  all  different 
from  those  of  the  first;  for  the  latter  leave  Xi  unchanged,  while  the 
former  do  not. 

We  select  now  any  substitution  t.^  which  converts  x^  into  x^  and 
form  for  the  third  linq  of  the  table 

^3  ,   S2'T3 ,   S3<T3  .   .   .  S„,<T^  . 

The  substitutions  of  this  line  may  then  be  shown  to  possess  proper- 
ties similar  to  those  of  the  second.  We  continue  in  this  way  until 
all  the  substitutions  of  the  group  are  arranged  in  n  lines  of  m  sub- 
stitutions each.     We  have  then 

Theorem  II.  If  the  number  of  sitbstitutions  of  a  transi- 
tive group,  which  leave  any  element  x^  unchanged,  is  m,  the  oi^der  r 
of  the  group  is  mn,  i.  e.,  always  a  multiple  of  n. 

The  following  extension  of  this  theorem  is  readily  obtained: 

Corollary.  If  x„,  Xi,,Xc,  .  .  .  are  any  arbitrary  elements  of 
the  group  G,  and  if  m  is  the  order  of  the  subgroup  of  G  which 
does  not  affect  these  elements,  then  the  order  of  G  is  mn',  where  n'  is 
the  number  of  distinct  systems  of  elements  Xa,  x^,  Xy,  .  .  .  by  tvhich 
the  substitutions  of  G  can  replace  x„,Xj,,x^,... 

§  62.  A  group  is  said  to  be  k-fold  transitive  when  its  substitu- 
tions permit  of  replacing  k  given  elements  by  any  k  arbitrary  ones. 
It  can  be  readily  shown  that  any  k  arbitrary  elements  can  then  be 
replaced  by  any  k  others.  The  definition  includes  the  case  where 
any  number  of  the  k  elements  remain  unchanged.  A  k-iold  transi- 
tive group  must  contain  one  or  more  substitutions  involving  any 
arbitrarily  chosen  cycle  of  the  A;*^  or  any  lower  order.  Thus  in  a 
four- fold  transitive  group  there  must  be  substitutions  which  leave 
x^  and  X2  unchanged  but  interchange  x^  and  0^45  and  which  are  there- 


GENERAL    CLASSIFICATION    OF    GEOUPS.  ^  73 

fore  of  the  form  (xi)  (X2)  (x^Xi)  .  .  .  The  same  group  must  also 
contain  a  substitution  which  leaves  x^,  x^,  x^,  x^  all  unchanged;  this 
may  of  course  be  the  identical  substitution. 

For  an  example  of  a  three-fold  transitive  group  we  may  take  the 
alternating  group  of  5  elements.  If,  for  instance,  we  require  a  sub- 
stitution which  leaves  x.2  unchanged  and  replaces  x^  and  x^  by  x^  and 
a?3  respectively,  the  circular  substitution  S  =  (xiX^Xs)  satisfies  these 
<;onditions,  and,  being  equivalent  to  the  two  transpositions  (x^Xr^)  (xiXs), 
belongs  to  the  alternating  group.  This  same  alternating  group  can- 
not however,  be  four- fold  transitive,  for  it  must  then  contain  a  sub- 
stitution which  converts  ccj,  X2,  x^,  x^  into  Xi,  X2,  x^,  Xr,  respectively; 
this  could  only  be  the  transposition  {x^x^,  and  this  cannot  occur  in 
the  alternating  group. 

In  general,  we  can  show  that  the  alternating  group  of  n  ele- 
ments is  always  (n  —  2)- fold  transitive.  The  requirement  that  any 
{n  —  2)  elements  shall  be  replaced  by  {n — 2)  others  may  take  any 
one  of  three  forms.  In  the  first  place  it  may  be  required  that 
(n — 2)  given  elements  shall  be  replaced  by  the  same  elements  in  a 
different  order,  so  that  two  elements  are  not  involved.  Secondly, 
the  requirement  may  involve  {n — 1)  elements,  or,  thirdly,  all  the  n 
elements. 

In  the  first  case  suppose  that  ^  is  a  substitution  which  satisfies 
the  conditions,  and  let  r  be  the  transposition  of  the  two  remaining 
elements.  Then  (tt  also  satisfies  the  conditions,  and  one  of  the  two 
substitutions  <t,  ar  belongs  to  the  alternating  group. 

If  {n  —  1)  elements  are  involved,  suppose  that  the  remaining 
element  is  x^ ,  so  that  neither  the  element  which  replaces  x,^  nor  that 
which  X,,  replaces  is  assigned.  The  elements  which  are  to  replace 
a?i,  a?2)  •  •  -Xn-\  are  all  known  with  the  exception  of  one.  Suppose 
that  it  is  not  known  which  element  replaces  x„_i.  Then  from  the 
elements  x^,  X2,  . .  .x^-i  we  can  construct  one  substitution  which 
satisfies  the  requirements,  say  (y—  {...  x^  x„  _-^  x^  ...)..  .  ,  and  from 
the  n  elements  a  second  one,  only  distinguished  from  the  first  in  the 
fact  that  x^_i  is  followed  by  x^,  thus  r  =  (.  .  .  a?«  cc„_ia?„a?^, .  .  .) 
=  (7 .  {Xf,x„).     Then  either  (t  or  r  belongs  to  the  alternating  group. 

Finally,  if  all  the  n  elements  are  involved,  there  are  two  elements 
for  which  the  substituted  elements  are  not  assigned.     Suppose  these 


74  THEOKY    OF    SUBSTITUTIONS. 

to  be  Xn-i  and  x^.  If  now  the  elements  are  arranged  in  cycles  in 
the  usual  manner,  there  will  be  two  cycles  which  are  not  closed,  the 
one  ending  with  a?«_i,  the  other  with  x,,.  We  can  then  construct 
two  substitutions  t  and  r  which  satisfy  the  requirements,  the  one 
being  obtained  by  simply  closing  the  two  incomplete  cycles,  the  other 
by  uniting  the  latter  in  a  single  parenthesis.  From  Chapter  II, 
Theorem  XI,  it  then  follows  that  either  ^r  or  r  belongs  to  the  alterna- 
ting group. 

The  alternating  group  of  n  elements  is  therefore  at  least  (n  —  2)- 
fold  transitive.  It  cannot  be  (n  —  1) -fold  transitive,  since  it  contains 
no  substitution  which  leaves  Xi,x.2,  .  .  .  a?„_2  unchanged,  and  con- 
verts x„_i  into  X,,. 

§  63.  If  G  is  a  A;-fold  transitive  group,  the  subgroup  G'  of  G 
which  does  not  affect  x^  will  be  (k — 1) -fold  transitive;  the  subgroup 
G"  of  G'  which  does  not  affect  X2  will  be  {k — 2)-fold  transitive, 
and  so  on.  Finally  the  subgroup  Gi^~'^^  which  does  not  affect 
Xi,X2,  .  .  .  Xk_i  will  be  simply  transitive.  Appljnng  Theorem  II 
successively  to  G^^~^\  .  .  .  G",  G',  G,  we  obtain 

Theorem  III.  The  order  r  of  a  k-fold  transitive  group  is 
equal  to  n{n  —  1)  Qn — 2)  .  .  .  {n — k-\-l)  m,  ivhere  m  is  the  order  of 
any  subgroup  ivhich  leaves  k  elements  unchanged. 

§  64.  A  simply  transitive  group  is  called  non-primitive  when 
its  elements  can  be  divided  into  systems,  each  including  the  same 
number,  such  that  every  substitution  of  the  group  replaces  all  the 
elements  of  any  system  either  by  the  elements  of  the  same  system 
or  by  those  of  another  system.  The  substitutions  of  the  group  can 
therefore  be  effected  by  first  interchanging  the  several  systems  as 
units,  and  then  interchanging  the  elements  within  each  separate  sys- 
tem. 

A  simply  transitive  group  which  does  not  possess  this  property^ 
is  called  primitive. 

For  example,  the  grdups 

Gi  =  |_1,  \X^x.2)j  (x^Xi),  (Xia?2)  (^3^4),  {XiX^}  [X2Xi),  [XiX^j  (^^2X3), 

\XiX^X2X^j^  [JXyiX^X2X^jj^ 
(t2  ^^^  I  i,  \XiX2X'^J^  {X^X^Xq)j   {X-jX^XqJj   \Xy)C^XopC2X^X-jX2,X^X^j  ^ 


GENERAL    CLASSIFICATION    OF    GROUPS.  75 

are  both  non-primitive.  G^  has  two  systems  of  elements,  x^ ,  x^  and 
Cfg,  x^\  G2  has  three  systems,  x^,X2,x^,     x^,  x^,  x^,  and  Xt,  x^,  Xg. 

The  powers  of  a  circular  substitution  of  prime  order  form  a 
primitive  group,  e.  g.  G^  —  [1,  {x^x^x^,  {x^x^x^l. 

The  powers  of  a  circular  substitution  of  composite  order 
form  a  non-primitive  group.  If  the  degree  of  the  substitution  is 
n  =pi'*i  .p2*2  .p^as  _  .  ^  where  Pi ,  ^2?  Ps?  •  •  •  are  the  different  prime 
factors  of  n,  the  corresponding  systems  of  elements  can  be  selected 
in  [(^tj  -j- 1)  [a^  + 1)  («3  -j- 1)  .  .  .  — 2]  different  ways,  as  is  readily  seen. 
For  example,  in  the  case  of  the  group 

G"4  =  |_1,  {x^x^XyXjX^Xo,)^  [XiX'jpCr^)  yx^x^x^,)^  {XiX^j  [x^Xr^)  [x^x^j^ 

[X-^XrflC;^)    [X2'^ffl^i)l     \XlX^XrflC^X^X.2) \^ 

we  may  take  either  two  systems  of  three  elements  each,  x^,X2,  x-^  and 
X2,  Xi,  Xq,  or  three  systems  of  two  elements  each,  x^Xt,  0^2,^5, 
and  x^,  Xq. 

A  theorem  applies  here,  the  proof  which  may  be  omitted  on 
account  of  its  obvious  character: 

Theorem  IV,  If,  for  a  non-primitive  group,  the  division 
of  the  elements  into  systems  is  possible  in  two  different  ways  such 
that  one  division  is  not  merely  a  subdivision  of  the  other,  then  a 
third  mode  of  division  can  also  be  obtained  by  combining  into  a 
new  system  the  elements  common  to  a  system  of  the  first  division 
and  one  of  the  second. 

It  must  be  observed  that  a  single  element  is  not  to  be  regarded 
as  a  "system''  in  the  present  sense.  Thus  the  group  (r^  above 
admits  of  only  two  kinds  of  systems. 

§  65.  The  elements  of  a  non-primitive  group  G  can  be  ar- 
ranged in  a  table,  as  follows.  The  first  line  contains  all  and  only 
those  substitutions 

Sj  =  1,  ^2,  S3,   .   .  .  S„j 

which  leave  the  several  systems  unchanged  as  units,  and  which 
accordingly  only  interchange  the  elements  within  the  systems.  (The 
line  will  of  course  vary  with  the  particular  distribution  of  the  ele- 
ments in  systems.) 

From  the  definition  of  transitivity,  (for  the  names  "primitive" 
and  "non-primitive"  apply  only  to  simply  transitive  groups),  there 


76  THEORY    OF    SUBSTITUTIONS. 

must  be  in  the  given  group  a  substitution  ^2  which  converts  any  ele- 
ment Xa  of  one  system  into  an  element  Xj,  of  another  system,  and 
which  consequently  interchanges  the  several  systems  in  a  certain 
way.     For  the  second  line  of  the  table  we  take 

We  show  then,  1)  that  all  the  substitutions  of  this  line  produce  the 
same  rearrangement  of  the  order  of  the  systems  as  (t^;  for  every  Sa 
leaves  this  order  unchanged;  2)  that  all  substitutions  which  produce 
the  same  rearrangement  of  the  systems  as  o-g  are  contained  in  this 
line;  for  if  r  is  one  of  these,  then  ra,,"^  =  S\,  so  that  r  =  s^(j2',  3) 
that  all  the  substitutions  of  the  second  line  are  different  from  one 
another;  and  4)  that  they  are  all  different  from  those  of  the  first 
line. 

If  there  is  then  still  another  substitution  ^3  which  produces  a 
new  arrangement  of  the  systems,  this  gives  rise  to  a  third  line  which 
possesses  similar  properties,  and  so  on. 

Tlieorem  V.  If  a  non-primitive  group  G  contains  a  sub- 
group Gi  of  order  m  ivhich  does  not  interchange  the  several  systems 
of  elements,  the  order  r  of  G  is  equal  to  mq,  where  q  is  a  divisor  of 
[J. ! ,  IX  being  the  number  of  systems. 

§  66.  If  we  denote  the  several  systems,  regarded,  so  to  speak, 
as  being  themselves  elements,  hj  A^,  A^,  .  .  .  A^,  then  all  the  substi- 
tutions of  any  one  line  of  the  table  above,  and  only  these,  produce 
the  same  rearrangement  of  the  Aj ,  A2 ,  .  .  .  A^ .  To  every  line  of  the 
table  corresponds  therefore  a  substitution  of  the  A's;  the  first  line, 
for  example,  corresponding  to  the  identical  substitution,  etc.  These 
new  substitutions  we  denote  by  §1  ~  1,  §25  •  •  •  ^a-  It  is  readily  seen 
that  they  form  a  new  group  (3.  For  the  successive  application  of 
§a  and  §(3  to  the  elements  A  produces  the  same  rearrangement  of 
these  elements  as  if  the  corresponding  ^a  and  <r^  were  successively 
applied  to  the  elements  x.  Accordingly,  since  o-a""^  =  S '  ^®  have 
also  §a§|3  — §y,  whereby  corresponds  to  the  line  of  the  table  above 
which  contains  ffy.  The  system  of  §'s  therefore  possesses  the  char- 
acteristic property  of  a  group. 

We  perceive  here  a  peculiar  relation  between  the  two  groups  G 
and  @.     To  every  substitution  s  of  the  former  corresponds  one  sub- 


GENERAL  CLASSIFICATION  OF  GROUPS.  77 

stitution  §>  of  the  latter,  and  again  to  every  §  of  @  corresponds  either 
one  substitution,  or  a  certain  constant  number  of  substitutions  s  of  G. 
And  this  correspondence  is  moreover  of  such  a  nature  that  to  the 
product  of  any  two  s's  corresponds  the  product  of  the  two  corres- 
ponding §'s. 

If  to  every  §  there  corresponds  only  one  s,  then  there  is  only  one 
substitution,  identity,  in  (S  which  leaves  the  order  of  the  systems  A 
unchanged.  The  two  following  groups  may  serve  as  an  example  of 
this  type.     Suppose  that 

(j-  =  |_1,  {X1X2)  yX^X^j  [X^Xq)^  (XiX^)  {X.2Xq)  \X^X^)J  [XiX^Xf,)  (X2'^b'^3)j 

i^XiX^)  (^X2X^)  (XqXq)^  yXiX^x^)  (^i^2'^3'^5JJ' 

Here  the  systems  A^,  Ao,  and  A^  are  composed  respectively  of  Xi 
and  X2,  x^  and  Xq,  and  Xi  and  Xr^.  The  corresponding  substitutions 
of  the  A^s  form  the  group 

(S  =  [1,  (A3A3),  (A,A2),  (A1A3A,),  (A1A3),  (AA2A3)]. 

§  67.    We  examine  more  closely  the  subgroup 

G-j  =  [_Si ,  §2 ,  S3 ,  .  .  .  s„  J 

of  the  group  G  oi  %  65.  Since  Gi  cannot  replace  any  element  of 
one  system  by  an  element  of  another  system,  it  follows  that  Gi  is 
intransitive.  Any  arbitrary  substitution  t  ot  G  transforms  G^  into 
t~^G^t=  G\.  The  latter  is  also  a  subgroup  of  G\  it  is  similar  to  G-^\ 
and  it  evidently  does  not  interchange  the  systems  of  G.  It  follows 
that  G\  =  G,. 

Suppose  that  any  system  of  a  non-primitive  group  consists  of  the 
elements  x\,  x'2,  x'^,.  .  .  The  subgroup  Gi  therefore  permutes  the 
elements  x'  among  themselves.  We  proceed  to  examine  whether 
these  elements  are  transitively  connected  with  one  another  by  the 
group  Gi ,  or  whether  this  is  the  case  only  when  substitutions  of  G 
are  added  which  interchange  the  systems  of  elements.  Suppose 
that  x\,x\j..  .  .  x^a,  and  again  a:^'a  +  ij  .  .  .  ^'s»  etc.,  are  transitively 
connected  by  (xj.  Then  G  contains  a  substitution  of  the  form 
t={x\x'a  +  i ...)...  f  and  since  t~^Git=  G^,  it  follows  that  t  re- 
places all  the  elements  a?'i ,  cc'2 >  •  •  •  ^'a  by  x'a+i .  .  .  x'p.  Further 
consideration  then  shows  that  x\ ,  x'2,  .  .  .  x'a  form  a  system  of  non- 
primitivity. 


78  THEORY    OF    SUBSTITUTIONS. 

Accordingly  if  the  systems  of  non-primitivity  are  chosen  at  the 
outset  as  small  as  possible,  then  the  group  G^  connects  all  the  ele- 
ments of  every  system  transitively. 

Assuming  the  systems  to  be  thus  chosen,  we  direct  our  attention 
to  those  cycles  of  the  several  substitutions  of  G^  which  interchange 
the  elements  x'^^x'^,  .  .  .  of  any  one  system  of  non-primitivity. 
These  form  a  transitive  group  H'.  Similarly  the  components  of  the 
several  substitutions  of  Gj  which  interchange  the  elements  a?/'*^  x.2^°-\ . . 
of  any  second  system  form  a  group  H^'^X  The  groups  H',  H",  .  .  . 
are  similar,  for  if  t  =  {x\x^^'^K  ..)...  is  a  substitution  of  G,  then 
the  transformation  t-^Git=  G^  will  convert  H'  into  IK'^X     The  order 

of  H'  is  a  multiple  of  —  and  a  divisor  of  — ^ ! ,  where  /f.  is  the  number 

//  /J. 

of  systems  of  non-primitivity. 

§  68.  The  following  easily  demonstrated  theorems  in  regard  to 
to  primitive  and  non-primitive  groups  may  be  added  here: 

Theorem  VI.  If  from  the  elements  x^^  x^.,  .  .  .  Xn  of  a  tran- 
sitive group  G  any  system  x\^  x'2,  .  .  .  can  be  selected  such  that 
every  substitution  of  G  ivhich  replaces  any  x'a  by  an  x'^  permutes 
the  x'^s  only  among  themselves,  then  G  is  a  non-primitive  group. 

Theorem  VII.  If  from  the  elements  x^,  Xo,  .  .  .  x„  of  a 
transitive  group  G  two  systems  x\,  x'2,  .  .  .  and  x'\,  x"^,  .  .  •  can  be 
selected  such  that  any  substitution  which  replaces  any  element  x'a 
by  an  x" ^  replaces  all  the  x'^s  by  a?"'s,  then  G  is  a  non-primitive 
group. 

Theorem  VIII.  Every  primitive  group  G  contains  substi- 
tutions iL'hich  replace  an  element  x'a  of  any  giveyi  system  x'^,  x'2,  .  .  . 
by  an  element  of  the  same  system,  and  ivhich  at  the  same  time  replace 
any  second  element  of  the  system  by  some  element  not  belongincj  to 
the  system.  * 

§  69.  The  preceding  discussion  has  led  us  to  two  general  prop- 
erties of  groups  which,  together  with  transitivity  and  primitivity, 
are  of  fundamental  importance. 

In§  67  the  subgroup  Gj  of  G  possessed  the  property  of  being 
reproduced  by  transformation  with  respect  to  every  substitution  t  of 

*Eudio:  Ueber  primitive  Grunpen.    Crelle  CI.  p.  l. 


GENEEAL    CLASSIFICATION    OF    GROUPS. 


79 


G,  so  that  for  every  t  we  have  t^^G^t—G^.  We  may  conveniently 
indicate  this  property  of  G^  by  the  equation 

G-'G,G=^G,    or    G,G=GG,. 

It  is  to  be  observed  however  that  this  notation  must  be  cautiously 
employed.  For  example,  if  G^  is  any  subgroup  of  G,  we  have 
always  Gi~'^GGi  =  G,  and  from  this  equation  would  apparently  fol- 
low GGi  —  GiG,  and  consequently  Gi  —  G~^  G^  G.  But  this  last 
equation  holds  only  for  a  special  type  of  subgroup  G^ .  The  reason 
for  this  apparent  inconsistency  lies  in  the  fact,  that  in  the  equation 
G~^  GiG=  G^  the  two  G's  represent  the  same  substitution  and  the 
two  G^i's  in  general  different  substitutions,  while  in  the  equation 
Gj~^  GGi  =  G  the  reverse  is  the  case. 

We  introduce  here  the  following  definitions. 

1)  Two  substitutions  s^  and  Sg  ct''^  comnnitatwe  *  if 

12  —  ^2    1  • 

2)  A  substitution  Sj  and  a  group  H  are  commutative  if 

s,H  =  Hs,. 

3)  Two  groups  H  and  G  are  commutative  if 

HG=GH. 

The  last  equation  is  to  be  understood  as  indicating  that  the  product 
of  any  substitution  of  H  into  any  substitution  of  G  is  equal  to  the 
product  of  some  substitution  of  G  into  some  substitution  of  H,  so 
that,  if  the  substitutions  of  G  are  denoted  by  s  and  those  of  H  by  t, 
then 

for  every  «  and  fi. 

Under  2)  s^  may  be  a  substitution  of  H-,  for  s^  and  H  are  then 
always  commutative.  Under  3)  a  case  of  special  importance  is  that, 
an  instance  of  which  we  have  just  considered,  for  which  if  is  a  sub- 
group of  G.  In  this  case  Sa  and  ss  of  the  equation  sj^  =  tySs  are 
always  to  be  taken  equal. 

A  subgroup  H  of  any  group  G,  for  which  G^^HG  =  i?,  is  called 
a  self -conjugate  subgroup  of  G. 

♦German:  "  vertauschbar";  French:  "exchangeable",  retained  as  "interchangeable  " 
by  Dolza:  Anier.  Jour.  Math.  XIII,  p.  11. 


80  THEORY    OF    SUBSTITUTIONS. 

The  following  may  serve  as  examples: 

1)  The  substitutions  s^  =  (ccia^g^a)  (^^^s^e)?  ^2  =  (xiXi)  {x2X^)  (x^x^) 
are  commutative;  for  their  product  is  (x^Xr^x^x^x^x^),  independently 
of  the  order  of  the  factors. 

Every  power  s«  of  any  substitution  s  is  commutative  with  every 
other  power  s^  of  the  same  substitution. 

Two  substitutions  which  have  no  common  element  are  commuta- 
tive. 

2)  The  group  H  =  [1,  {x,x,)  (x^x^),  (x^x^)  (x^x^),  (x,Xi)  (x^x^)']  is 
commutative  with  every  substitution  of  the  four  elements  x^,  X2, 
x^j  x^. 

The  alternating  group  of  n  elements  is  commutative  with  every 
substitution  of  the  same  elements. 

3)  The  group  H  of  2),  being  commutative  with  the  symmetric 
group  of  the  four  elements  x^,  X2,  x<i,Xi,  is  a  self -conjugate  sub- 
group of  the  latter. 

The  alternating  group  of  n  elements  is  a  self -conjugate  sub- 
group of  the  corresponding  symmetric  group. 

Every  group  G  of  order  r,  which  is  not  contained  in  the  alterna- 
ting group  Aj  contains  as  a  self- conjugate  subgroup  the  group  H  of 
order  ^r  composed  of  those  substitutions  of  G  which  are  contained 
in  A  (Theorem  YIII,  Chapter  II). 

The  identical  substitution  is,  by  itself,  a  self-conjugate  sub- 
group of  every  group. 

§  70.  We  may  employ  the  principle  of  commutativity  to  further 
the  solution  of  the  problem  of  the  construction  of  groups  begun  in 
Chapter  II  (§§  33-40). 

All  substitutions  of  n  elements  which  are  commutative  with  any 
given  substitution  of  the  same  elements,  form  a  group. 

For  if  #1 ,  ^2  •  •  •  are  commutative  with  s,  it  follows  from 

1  1  —     )        ^2        ^^2  —  "? 

that 

{tA)-'s{t,t2)  =  s/ 

so  that  the  product  t^t^  also  occurs  among  the  substitutions  t. 

All  substitutions  of  n  elem^ents  which  are  commutative  with  a 


GENEEAL    CLASSIFICATION    OF    GROUPS.  81 

given  group  G  of  the  same  elements  form  a  group  which  contains  G 
as  a  self-conjugate  subgroup. 

For  from 

tr'Gt,  =  G,     t,-'Gt2  =  G, 
follows 

(t,t,)-'G{t,t,)  =  G', 

afid  among  the  f  s  are  included  all  the  substitutions  of  G, 

If  two  commutative  groups  G  and  H  have  no  substitution,  e^^^ept 
the  identical  substitution,  in  common,  then  the  order  of  the  smallest 

group 

K=\G,H\ 

is  equal  to  the  product  of  the  orders  of  G  and  H. 

§  71.  If  a  group  G  of  order  2r  contains  a  subgroup  H  of 
of  order  r,  then  H  is  a  self- conjugate  subgroup  of  G. 

For  if  the  substitutions  of  H  are  denoted  hj  1,  s^,  s^,  .  .  .  s,. ,  and 
if  t  is  any  substitution  of  G  which  is  not  contained  in  H,  then 
t,  ts2,  ts^,  .  .  .  ts,.  are  the  remaining  r  substitutions  of  G.  But  in  the 
same  way,  f,  Sg^?  ^3^,  .  .  .  s,.t  are  also  these  remaining  substitutions. 
Consequently  every  substitution  Sat  is  equal  to  some  tsp,  that  is,  wr 
have  in  every  case  t~^s^t  =  s^,  and  therefore  G~^HG  =  H. 

If  a  group  G  contains  a  self-conjugate  subgroup  H  and  any 
other  subgroup  K,  then  the  greatest  subgroup  L  common  to  H  and 
K  is  a  self-conjugate  subgroup  of  K.     If  the  orders  of  G,  H,  K,  L 

are  respectively  g,  h,  k,  Z,  then  —  is  a  multiple  of  — . 

For  if  s  is  any  substitution  of  K,  then  s~^Z<s  is  contained  in  K, 
since  all  the  separate  factors  s  ~  \  L,  s  are  contained  in  K.  But 
s~^Ls  is  also  contained  in  H,  for  L  is  a  subgroup  of  H  and 
s~^Hs  =  II.  Consequently  s~^Ls  is  contained  in  L,  and,  as  these 
two  groups  have  the  same  number  of  substitutions,  we  must  have 
s~^Ls  —  L,  and  L  is  a  self- conjugate  subgroup  of  K. 

The  relation  between  the  orders  of  the  four  groups  follows  at 

once  from  the  formula  of   Frobenius  (§  48).     We  have  only  to  take 

for  the  K  of  this  formula  the  present  group  H,  and  to  put  all  the 

n       TYh 
di,d2,  .  .  .d„,  equal  to  I.    We  have  then  tt  =  -y  • 


82  THEORY    OF    SUBSTITUTIONS. 

A  self- conjugate  subgroup  of  a  transitive  group  either  affects 
every  element  of  the  latter,  or  else  it  consists  of  the  identical  substi- 
tution alone. 

For  if  H=  G~^HG  is  a  self- conjugate  subgroup  of  the  transi- 
tive group  G,  and  if  H  does  not  affect  the  element  x^ ,  then,  since  G 
contains  a  substitution  S\  which  replaces  x^  by  cfx,  it  would  follow 
that  S},~^Hs;^  =  H  would  also  not  affect  a^x,  that  is,  that  jff  would 
not  affect  any  element. 

If  a  self-conjugate  subgroup  of  a  transitive  group  G  is  intransi- 
tive,, then  G  is  non-primitive  and  H  only  interchanges  the  elements 
within  the  several  systems  of  non-primitivity. 

For  suppose  that  x^  and  Xx  belong  to  two  different  systems  of 
intransitivity  with  respect  to  H.  Then  G  contains  a  substitution  S\ 
which  replaces  x^  by  x^,  and  since  S\~^Hsx  =  H,  it  follows  that 
S\~^HS),  must  replace  X\  only  by  elements  transitively  connected 
with  X;^  with  respect  to  H.  But  Sa~^  replaces  a?x  by  x^  and  H  re- 
places x^  by  every  element  of  the  same  system  of  intransitivity  with 
Xi .  Consequently  the  remaining  factor  s^  must  replace  every  ele- 
ment of  the  system  containing  x^  by  an  element  of  the  system  con- 
taining a?A.  The  systems  of  intransitivity  of  H  are  therefore  the 
systems  of  non-primitivity  of  G. 

§  72.  Another  important  property  is  that  of  the  correspond- 
ence of  two  groups,  of  which  an  instance  has  already  been  met 
with  in  §  66.  The  two  groups  G  and  @  of  this  Section  were  so 
related  that  to  every  substitution  s  of.  G  corresponded  one  substitu- 
tion §  of  @,  and  to  every  §  corresponded  a  certain  number  of  s's. 
The  correspondence  was  moreover  such  that  to  the  product  of  any 
two  s's  corresponded  the  product  of  two  corresponding  §'s. 

We  may  consider  at  once  the  more  general  type  of  correspond- 
ence, *  where  to  every  substitution  of  either  group  correspond  a 
certain  number  of  substitutions  of  the  other,  and  to  every  pro- 
duct SaSp  corresponds  every  product  §a^^  of  corresponding  §'s  and 
vice  versa.  We  may  then  readily  show  that  to  every  substitution  of 
the  one  group  correspond  the  same  number  of  substitutions  of  the 
other.     For  if  to  1  of  the  group  G  correspond  1,  §2?  ^3j  .  .  .  §<^  of  (51, 

*  A.  Capelli :  Battaglini  Gior,  1878,  p.  32  seq. 


GENEEAL    CLASSIFICATION    OF    GEOUPS.  83 

then,  if  §  corresponds  to  s,  all  the  substitutions  §,  §§2 ,  §§3 ,  .  .  .  ^% 
correspond  to  s,  by  definition.  Conversely,  if  any  substitution  §' 
corresponds  to  s,  then  § ~ ^ §'  corresponds  tos~^s  =  l,  and  therefore 
§"^§'  is  contained  in  the  series  1,  §9?  §3?  •  .  •  §«•  Consequently  the 
series  §,  §§2>  ^^35  •  •  •  ^%  contains  all  the  substitutions  of  @  which  cor- 
respond to  s,  and  the  number  q  is  constant  for  every  s.  Similarly 
to  every  §  correspond  the  same  number  p  of  the  substitutions  s. 
It  is  evident  at  once  that 

the  substitutions  of  G  ((5))  which  correspond  to  the  identical  substi- 
tution of  @  (G)  fonn  a  group  H  (§)  which  is  a  self- conjugate  sub- 
group of  G  ((S). 

The  correspondence  of  two  groups  as  just  defined  is  called  iso- 
morphism.    If  to  every  substitution  of  G  correspond  q  substitutions 

of  ®,  and  to  every  substitution  of  @  p  substitutions  of  (7,  then  G 

» 

and  (3  are  said  to  be  (p-q)-fold  isomorphic,  or  if  p  and  q  are  not 
specified,  manifold  isomorphic.  If  p  =  q  =  l,  the  groups  are  said 
to  be  simply  isomorphic.  * 

Examples. 

I.  The  groups 

Cr  —  \_1,  \X1X2)  {X^Xq)  [X^Xr,),  yXiXcij  yXoXr,)  (a^^iCgj,  {XiX^j  [X^Xf^j  {X-^Xr,), 

yXiX^x^)  yX2X^x^),  yXiX^Xr^)  {X2X^Xq)j, 

^    ==  L-'-J  (^1^2)5    (m^s))    (^2^3))    (^1^2^3)?    (^1^3^2)J 

are  simply  isomorphic,  the  substitutions  corresponding  in  the  order 
as  written.  For  if  any  two  substitutions  of  G,  and  the  corres- 
ponding substitutions  of  T,  are  multiplied  together,  the  resulting 
products  again  occupy  the  same  positions  in  their  respective  groups. 

II.  The  groups 

G  =  [1,  (x,x2)l     r  =  [1,  {-A)  {^A\  i^A)  (4^2?.),  (^1^.)  (^2^3)] 

are  (1-2) -fold  isomorphic.  Corresponding  to  1  of  (r-we  may  take, 
beside  1,  any  other  arbitrary  substitution  of  /'.  It  follows  that  F  is 
simply  isomorphic  with  itself  in  different  ways. 

*  C/.  Camille  Jordan :  Traite  etc.,  §  67-74,  where  the  names  "  holoedric  "  and  *nierl- 
edric"  isomorphism  are  employed.  These  have  been  retained  by  Dolza:  Amer.Jour, 
Vol.  XIII.  The  "simply,  manifold,  (p-Q)-fold  isomorphic  "  above  rei)resent  the  "ein- 
stufig,  mehrstuflg,  (p-g)-stufig  isomorph  "  of  the  (Jerman  edition. 


84  THEOBY    OF    SUBSTITUTIONS. 

III.     The  groups 

Cr  =  l^i,  {X1X2)  \X^X^),  {X1X2)  \X2X^)J  \^iXi)  \X2X^)_\, 

r=  [1,  (?.f,f,),  (f,?,?,),  (?,?,),  (f,e,),  (^A)} 

are  (2-3)-fold  isomorphic.  To  the  substitution  1  of  6r  correspond 
1,  (ci^2^3))  (^i^3^2)  of  ^5  and  conversely  to  1  of  F  correspond 
1,  (x^x^)  (x^x^)   of  G. 

§  73.  If  G  and  F  are  (m-n)-fold  isomorphic,  then  their  orders 
are  in  the  ratio  of  m:n. 

If  L  is  a  self-conjugate  subgroup  of  G,  and  if  A  is  the  corres- 
ponding subgroup  of  T,  then  A  is  a  self- conjugate  subgroup  of  F. 

For  from  G'^LG  —  L  follows  at  once  F-^AF=  A.  In  the  case 
of  (p-l)-fold  isomorphism,  it  may  however  happen  that  the  group 
A  consists  of  the  identical  substitution  alone. 

§  74.  Having  now  discussed  the  more  elementary  properties  of 
groups  in  reference  to  transitivity,  primitivity,  commutativity,  and 
isomorphism,  we  turn  next  to  certain  more  elaborate  investigations 
devoted  to  the  same  subjects. 

The  m  substitutions  of  a  transitive  group  G  which  do  not  affect 
the  element  Xi  form  a  subgroup  G^  of  G.  Similarly  the  substitutions 
of  G  which  do  not  affect  X2  from  a  second  subgroup  G2,  and  so  on 
to  the  subgroup  G,,  which  does  not  affect  x„.  All  these  subgroups 
are  similar;  for  if  <Ta  is  any  substitution  of  G  which  replaces  x^  by 
Xaj  we  have  (^a~^Gi<Ta— G2.  The  groups  Ga  are  therefore  all  of 
order  m. 

If  now  we  denote  by  [g]  the  number  of  those  substitutions  of  Gy^ 
which  affect  exactly  g  elements  but  leave  the  remaining  {n  —  q — 1) 
unchanged,  then  [q\  is  also  the  corresponding  number  for  each 
of  the  other  groups  G2-,G^,  .  .  .  G,,.  It  follows  then  from  the  mean- 
ing of  the  symbol  [q]  that 

m  =  [»-l]  +  [«-2]  +  . . .  +[g]+  . . .  +[2]  +  [0], 

where  the  symbol  [1]  does  not  of  course  occur,  and  [0]  =  1. 
6ri,  (t2,  .  .  .  Gn  therefore  possess  together  n\n  —  1]  substitutions 
which  affect  exactly  (ri — 1)  elements.  These  are  all  different,  for 
any  substitution  which  leaves  only  Xa  unchanged  occurs  in  Ga,  but 
cannot  also  occur  in  G^.     But  this  is  not  the  case  with  substitutions 


GENEEAL    CLASSIFICATION    OF    GROUPS.  85 

which  affect  exactly  (n  —  2)  elements;  for  if  any  one  of  these  leaves 
both  Xa  and  Xp  unchanged,  it  will  occur  in  both  Ga  and  Gp .  Accord- 
ingly every  one  of  these  n\n  —  2]  substitutions  is  counted  twice, 
and  G  therefore  contains  ^n[n  —  2]  substitutions  which  affect 
exactly  (n — 2)  elements.  Similarly  every  one  of  the  n[q]  substitu- 
tions of  g  elements  which  occur  in  G^,  G^,  .  .  .  G^  is  counted  (n — q) 

71/ 

times,  and  there  are  therefore  only [g]  different  substitutions 

in  G  which  affect  exactly  q  elements.  We  have  then  for  the  total 
number  of  substitutions  in  G,  which  affect  less  than  n  elements 

If  this  number  is  subtracted  from  that  of  all  the  substitutions  in  G, 
the  remainder  gives  the  number  of  substitutions  in  G  which  affect 
exactly  7i  elements.     But  from  Theorem  II 

r  ==  mn  —  n[n  —  1]  +  n\n —  2]  +  .  .  .  +  n^q']  +  .  .  .  +  w[0]-, 

and  consequently  the  required  difference  A^  is 

4[»-2]+|[«-3]+...  +  ^^=^M  +  ...+!^[0]). 
No  term  in  the  parenthesis  is  negative.  The  last  one  is  equal  to 
since  [0]  =  1.     Consequently  N>^{n — 1). 

Theorem  IX.  Every  transitive  gromp  contains  at  least 
(n — 1)  substitutions  which  affect  all  the  n  elements.  If  there  are 
more  than  {n — 1)  of  these,  then  the  group  also  contains  substitu- 
tions which  affect  less  than  (n  —  1)  elements.  * 

Corollary.  A  k-fold  transitive  group  contains  substitutions 
which  affect  exactly  n  elements,  and  others  which  affect  exactly 
{n — 1),  {n  —  2),  .  .  .  {n  —  k-\-l)  elements. 

Those  substitutions  which  affect  exactly  k  elements  we  shall  call 
substitutions  of  the  fc*'^  class.  We  have  just  demonstrated  the 
existence  of  substitutions  of  the  n^^,  or  highest  class. 

If  we  consider  a  non- primitive  group  G,  there  is  (§  66)  a  second 
group  @  isomorphic  with  G,  the  substitutions  of  which  interchange 

♦C.  Jordan:  Liouville  Jour.  (2),  XVII,  p  351. 


86  THEORY   OF   SUBSTITUTIONS. 

the  elements  Ai,  A^,  . .  ,  A^  exactly  as  the  corresponding  substitu- 
tions of  G  interchange  the  several  systems  of  non-primitivity. 
Since  G  is  transitive,  ©  is  also  transitive.  From  Theorem  IX  fol- 
lows therefore 

Theorem  X.  Every  non-primitive  group  G  contains  substi- 
tutions which  interchange  all  the  systems  of  non-primitivity. 

§  75.  We  construct  within  the  transitive  group  G  the  subgroup 
H  of  lowest  order,  which  contains  all  the  substitutions  of  the  high- 
est class  in  G,  and  prove  that  this  group  H  is  also  transitive. 

H  is  evidently  a  self- conjugate  subgroup  of  G.  If  H  were 
intransitive,  G  must  then  be  non-primitive  (Theorem  VI).  If  this 
is  the  case,  let  @  be  the  group  of  §  66  which  affects  the  systems 
Ai ,  Ag ,  . . .  Afj,  regarded  as  elements.  @  is  transitive.  To  substitu- 
tions of  the  highest  class  in  @  correspond  substitutions  of  the  high- 
est class  in  G.  (The  converse  is  not  necessarily  true).  Suppose  that 
^  is  the  subgroup  of  the  lowest  order  which  contains  all  the  substi- 
tutions of  the  highest  class  in  @.'  To  §  then  corresponds  either  Hot 
a  subgroup  of  H.  If  $  is  transitive  in  the  A's,  H  is  transitive  in  the 
x's.  The  question  therefore  reduces  to  the  consideration  of  the 
groups  @  and  §.  §  can  be  intransitive  only  if  @  is  non-primitive 
and  G^  accordingly  contains  more  comprehensive  systems  of  non- 
primitivity.  If  this  were  the  case,  we  should  again  start  out  in  the 
same  way  from  @  and  ^,  and  continue  until  we  arrive  at  a  primitive 
group.     The  proof  is  then  complete. 

Theorem  XI.  In  every  transitive  group  the  substitutions  of 
the  highest  class  form  by  themselves  a  transitive  system. 

§  76.  Suppose  a  second  transitive  group  G'  to  have  all  its  sub- 
stitutions of  the  highest  class  in  common  with  G  of  the  preceding 
Section.  If  then  we  construct  the  subgroup  H'  for  G',  correspond- 
ing to  the  subgroup  H  of  G,  we  have  H'  =  H. 

Moreover  the  number  iVj  of  the  substitutions  of  the  highest  class 
in  H  is 

<|[n-2],  +  |[«-3J,  +  ..+^^^M.+  ..+^[0]).> 

where  [g]i  has  the  same  relation  to  H  as  [q]  to  G.  But  the  number 
iV"i  is,  as  we  have  just  seen  equal  to  the  iV  of  §  74.     Consequently 


GENERAL    CLASSIFICATION    OF    GROUPS.  87 

«H([»-2]-[»-2],)  +  S([»-3]-[«-3]0+  . . . 

•••+^^^=^(M-Mi)+---f=o. 

But,  since  H  is  entirely  contained  in  G,  it  follows  that  [gJ^Mi, 
and  therefore  the  left  hand  member  of  the  equation  above  can  only- 
vanish  if  each  parenthesis  is  0.  Consequently  G  and  G'  can  only 
difiPer  in  respect  to  substitutions  of  the  (n  —  1)*^  class. 

Theorem  XII.  If  two  transitive  groups  have  all  their  sub- 
stitutioiis  of  the  highest  class  in  common^  they  can  only  differ  in 
those  substitutions  which  leave  only  one  element  unchanged. 

§  77.  Let  G  be  any  transitive  group  and  Gi^Gi,  . . .  Gn  those 
subgroups  of  G  which  do  not  affect  ccj,  a?2, . . .  a?„  respectively. 
These  groups  are,  as  we  have  seen,  all  similar.  If  now  G^ ,  and  con- 
sequently (t2,  . .  .  (r„,  are  A;-fold  transitive,  then  G  is  at  least  (k -{-!)- 
fold  transitive.  For  if  it  be  required  that  the  {k  -f- 1)  elements 
Xi,X2i .  .  .Xj,^^  shall  be  replaced  by  Xi^,  .t.^?  •  •  •  ^^k+i  respectively,  we 
can  find  in  G some  substitution  s  which  replaces  Xi,X2,x^,  . .  .Xf,^i 
by  a?,i,a?;,„,a?;,3,. .  .a?;,^^j,  where  Xj,^,Xh^,...  may  be  any  elements 
whatever.  Again  Gi^  contains  some  substitution  t  which  replaces 
^hoi  ^/igj  ...  by  a?,2)^»3j  •  •  •  Consequently  the  substitution  st  ot 
G  satisfies  the  requirement. 

From  this  follows  the  more  general 

Theorem  XIII.  //  a  group  G  is  at  least  k-fold  transi- 
tive, and  if  the  subgroup  of  G  which  leaves  k  given  elements  un- 
changed is  still  h-fold  transitive,  then  G  is  at  least  {k-\-h)-fold  tran- 
sitive.^ 

§  78.  Suppose  that  those  substitutions  of  a  A;- fold  transitive 
group  (t,  which,  excluding  the  identical  substitution,  affect  the 
smallest  number  of  elements,  are  of  the  q^^  class,  i.  e.,  that  they 
affect  exactly  q  elements.  The  question  arises  whether  there  is  any 
connection  between  the  numbers  k  and  q. 

In  the  first  place  suppose  k^q,  and  let  one  of  the  substitu- 
tions of  the  q^^  class  contained  in  (7  be  s  =  (o^jCCg ...)...(...  a^g_  i  x.^). 
Then,  on  account  of  its  fc-fold  transitivity,  G  also  contains  a  substitu- 

*G.  Frobenius:  Ueber  die  Congruenz  nach  einem  aus  zwei  endlichen  Gruppen  geb- 
ildeten  Doppelmodul.    Crelle  CI.  p.  290. 


88  THEORY    OF    SUBSTITUTIONS. 

tion  (T,  which  replaces  a?i ,  a?2 » •  •  •  ^2-1  ?  x^byx^,  X2,...  Xq_i,  x^  (x >  q), 
and  which  is  therefore  of  the  form 

(T  —  (xi)  (x^)  ...{Xg_,)  (x^x^  ...). 
We  have  then 

(fT-'s(T)s-'  =  l{x,x, ...)...(...  x^_,x^)']s-'  =  (x^^.x^Xq), 

and  since  this  substitution  affects  only  3  elements,  it  follows  that 
^<3. 

Secondly,  suppose  k  <q.  The  substitutions  of  the  q^^  class  may 
then  be  of  either  of  the  forms 

Si  =  {X1X2  ...),..{...  X^._iX}t.  ...)..  .y.  .  .  Xq), 

§2  =   (^1^2  •   •   •  )  •  A  •  (  •   •   •  ^k—l)  {^k  ...)...(...  Xq). 

In  the  first  case  we  take 

<Ti  =  (xi)  (.T2)  .  •  •  {00k-i)(XkX^.  ..)     (A;  +  l)£^ig, 
and  in  the  second 

^2  =  (^1)  (^2) . . .  i^'k-i)  (^A-a^A . . . )    /  >  g. 
It   is  evident    that  both   are    possible,   if  in  the  latter  case  it  is 
remembered  that  n>  q.     We  obtain  then 

(Ti        Siffi  =  {X^X2  ...)...(...  Xk_iXif  .   .  .  ), 
(T^-'^S^ff^  =  (X^X2  ...)•••(•••  ^A—l)  (a^A  •••  )» 

and  if  we  form  now  {frr^Si(T^)Si~\  the  first  {k — 2)  elements  are 
removed,  and  there  remain,  at  the  most,  q-\-(q — k)  —  (A; — 2) 
=  2q  —  2A;  +  2.  Similarly,  if  we  form  {(Tf  ^82(72)82"  \  the  first  (A:  +  1) 
elements  are  removed,  and  there  remain,  at  the  most,  q-\-{q — k-{-l) 
—  (A; — l)  =  2g — 2A;  +  2.  By  hypothesis,  this  number  cannot  be 
less  than  q.     Consequently 

g>2A;  — 2. 

Theorem  XIV.  If  a  k-fold  transitive  group  contains  any 
substitution,  except  the  identical  substitution,  which  affects  less 
than  (2k — 2)  elements,  it  contains  also  substitutions  which  affect  at 
the  most  only  three  elements. 

This  theorem  gives  a  positive  result  only  if  A;  >  2.  In  this  case, 
by  anticipating  the  conclusions  of  the  next  Section,  we  can  add  the 
following 


GENERAL    CLASSIFICATION    OF    GROUPS.  89 

Corollary.  //  a  k-f old  transitive  group  ky2  contains  sub- 
stitutio7is,  different  from  identity,  which  affect  not  more  than 
(2k — 2)  elements,  it  is  either  the  alternating  or  the  symmetric  group. 

We  may  now  combine  this  result  with  the  corollary  of  Theorem 
IX.  If  G  is  A;- fold  transitive,  it  contains  substitutions  of  the  class 
{n — k-\rV).  Accordino^ly  q^{n — k-\-l).  If  G  is  neither  the 
alternating  nor   the  symmetric  group,  q>{2k  —  2).     Consequently 

(n  — /j+1)  >  (2/^— 2)  and  A;£^^i^. 

o 

Theorem  XV.  If  a  group  of  degree  n  is  neither  the  alter- 
nating nor  the  symmetric  group,  it  is,  at  the  most,  I  —  -f- 1  \- fold  tran- 
sitive. 

That  the  upper  limit  of  transitivity  here  assigned  may  actually 
occur  is  demonstrated  by  the  five-fold  transitive  group  of  twelve 
elements  discovered  by  Matthieu, 

(y.x)  {x,x,)  (a-gic^)  (XiX^),  (2/22/1)  (xiX;)  (x,x^)  (x^x,), 

(y^Vi)  (^1^5)  (-^s-^t)  (^'4^6),  (2/42/3)  (^1^3)  (^4^-5)  {^o^i)\- 

§  79.  Theorem  XVI.  //  a  k-f  old  transitive  group  {k>l) 
contains  a  circular  substitidion  of  three  elements,  it  contains  the 
alternating  group. 

Suppose  that  s  =  (^x-iX^x.^)  occurs  in  the  given  group  G.  Then, 
since  G  is  at  least  two- fold  transitive,  it  must  contain  a  substitution 
(T  =  (x-i)  {xiX^X\ .,.)...  and  consequently  also 

r  —  fT  -  's^  =  (x-^XiXfj),         r~^sr  =■  [XiXiX^). 

In  the  same  way  it  appears  that  G  contains 

{XiX2X^),  {X^X^X^),  .  .  . 

Consequently  (§  35)  G  contains  the  alternating  group. 

Theorem  XVII.  If  a  kfold  transitive  group  (k  >  1)  con- 
tains a  transposition,  the  group  is  symmetric. 

The  proof  is  exactly  analogous  to  the  preceding. 
For  simply  transitive  groups  the  l^st  two  theorems   bold  only 
6a 


90  THEORY    OF    SUBSTITUTIONS. 

under  certain  limitations,  as  appear  from  the  following  instances 

(xi  =  [1,  (a?ia?2),  {x^x^)y  {X1X2)  {x^x^},  (^1^3)  (^2^4)?  (^1^4)  (^2^3)? 

\XiXsX2X^),  [XiX^X2X^)^j 
1X2  ::^=  ^  Ij  [XiX2X^jf  [X^XqXqJj  yX-jX^Xg)j  yXiX^XgX^XQX'jX^X^X^f  ^ . 

Both  of  these  are  transitive.  But  the  former  contains  a  substi- 
tution of  two  elements,  without  being  symmetric,  and  the  latter  a 
substitution  of  three  elements  without  being  the  alternating  group. 

§  80.  An  explanation  of  this  exception  in  the  case  of  simply 
transitive  groups  is  obtained  from  the  following  considerations. 

If  we  arbitrarily  select  two  or  more  substitutions  of  n  elements, 
it  is  to  be  regarded  as  extremely  probable  that  the  group  of  lowest 
order  which  contains  these  is  the  symmetric  group,  or  at  least  the 
alternating  group.  In  the  case  of  two  substitutions  the  probability 
in  favor  of  the  symmetric  group  may  be  taken  as  about  |,  and  in 
favor  of  the  alternating,  but  not  symmetric,  group  as  about  J. 
In  order  that  any  given  substitutions  may  generate  a  group  which 
is  only  a  part  of  the  n\  possible  substitutions,  very  special  relations 
are  necessary,  and  it  is  highly  improbable  that  arbitrarily  chosen 

substitutions  s,:  =  \  J  J  '  '  \S  I  should  satisfy  these  conditions.  The 

exception  most  likely  to  occur  would  be  that  all  the  given  substitu- 
tions were  severally  equivalent  to  an  even  number  of  transposi- 
tions and  would  consequently  generate  the  alternating  group. 

In  general,  therefore,  we  must  regard  every  transitive  group 
which  is  neither  symmetric  nor  alternating,  and  every  intransitive 
group  which  is  not  made  up  of  symmetric  or  alternating  parts,  as  de- 
cidedly exceptional.  And  we  shall  expect  to  find  in  such  cases 
special  relations  among  the  substitutions  of  the  group,  of  such  a 
nature  as  to  limit  the  number  of  their  distinct  combinations. 

Such  relations  occur  in  the  case  of  the  two  groups  cited  above. 
Both  of  them  belong  to  the  groups  which  we  have  designated  as 
non-primitive.  In  G^  the  elements  x^ ,  X2  form  one  system,  and 
a?3,  a?4  another;  it  is  therefore  impossible  that  G^  should  include,  for 
example,  the  transposition  {x^x^.  In  G2  there  are  three  systems  of 
non-primitivity  x^,X2,x^^  Xi,Xr^,x^,  and  x-i,x^,x^,  G^^  therefore 
cannot  contain  the  substitution  (xiX^x-!). 


GENERAL    CLASSIFICATION    OF    GROUPS.  .         91 

It  is,  then,  evidently  of  importance  to  examine  the  influence  of 
primitivity  on  the  character  of  a  transitive  group,  and  we  turn  our 
attention  now  in  this  direction. 

§  81.     With  the  last  two  theorems  belongs  naturally 

Theorem  XVIII.  If  a  primitive  group  contains  either  of 
the  two  substitutions 

it  contains  in  the  former  case  the  alternating,  in  the  latter  the  sym- 
metric group. 

The  proofs  in  the  two  cases  are  of  the  same  character.  We  give 
only  that  for  the  latter  case. 

From  Theorem  VIII,  the  given  group  must  contain  a  substitu- 
tion which  leaves  x^  unchanged  and  replaces  X2  by  a  new  element 
X3,  or  which  leaves  X2  unchanged  and  replaces  x^  by  a  new  element 
x^ ,  or  which  replaces  x^  by  X2  or  X2  by  x^  and  the  latter  element  in 
either  case  by  a  new  element  x-^.  If  then  we  transform  r  with 
respect  to  this  substitution,  we  obtain  a  transposition  r'  connecting 
either  x^  or  X2  with  x^ ,  for  example  r'  =  (cCjCCg).  The  presence  of  r 
and  r'  in  the  group  shows  that  the  latter  must  contain  the  symmet- 
ric group  of  the  three  elements  x^,  X2,x^.  From  Theorem  VIII 
there  must  also  be  in  the  given  group  a  second  substitution  which 
replaces  one  of  these  three  elements  by  either  itself  or  a  second  one 
among  them,  and  which  also  replaces  one  of  them  by  a  new  element 
x^.     Suppose  this  substitution  to  be,  for  instance, 

S  —  I  .  .  .  t4v2t*^i  .  .  •  X^Xi^  ..,).,, 

We  obtain  then 

t       o  1*^2      3/         \      1      4/5  \ 

and  it  follows  that  the  given  group  contains  the  symmetric  group  of 
the  four  elements  x^^x^^x^,  CC4;  and  so  on.    • 

§  82.     We  can  generalize  the  last  theorem  as  follows: 

Theorem  XIX.     If  a  primitive  group  G  with  the  elements 

Xi,  X2 .  .  .  Xn  contains  a  primitive  subgroup  H  of  degree  k  <Cn,  then 
G  contains  a  series  of  primitive  subgroups  similar  to  H, 

Hii  II2,  Ha,  '  '     IIn-k  + 


92  THEOBY    OF    SUBSTITUTIONS. 

such  that  every  Hx  affects  the  elements  a?i ,  cTo  5  •  •  •  ^;i  -  n  ^7.  +  a  -  n 
where  Xi,X2,  .  .  .Xk_i  may  be  selected  arbitrarily. 

We  take  H^  =  H  and  transform  H  with  respect  to  all  the  substi- 
tutions of  G  into  H^ ,  H\ ,  H'\ ,  .  .  .  Now  let  H\  be  that  one  of  the 
transformed  groups  which  connects  the  k  elements  x-^,  x.^  .  . ,  ocj,  of 
-H"i  with  other  elements,  but  with  the  smallest  number  of  these. 
We  maintain  that  this  smallest  number  is  one.  For  if  several  new 
elements  l^i,  l2»  •  •  •  occurred  in  H\^  then  from  Theorem  VIII  there 
must  be  in  the  primitive  group  H\  a  substitution  which  replaces 
one  ^  by  another  c  and  at  the  same  time  replaces  a  second  ^  by  one 
of  the  elements  Xi^X2,  . . .  Xj,.     Suppose  that 

is  such  a  substitution,  the  case  where  1^  =  y  being  included.  Then 
H'\  —  tH^t~^  will  still  contain  c^  but  will  not  contain  ?„•  H'\ ,  there- 
fore contains  fewer  new  elements  c  than  H\ .  Consequently  if  H'^ 
is  properly  chosen,  it  will  contain  only  one  new  element,  say  .T;,  +  i 
It  will  therefore  not  contain  some  one  of  the  elements  of  H^ ,  say  Xa^ 
We  select  then  from  H^  a  substitution  u  =  (.  .  .  XaX^. ...)...  and 
form  the  group  u~^H\u  =  H2.     This  group  contains 

Xi,  X2  .  .  .  £*?A-  —  n  ^A-  +  n 

but  not  x^.  In  the  same  way  we  can  form  a  group  H.  which  affects 
only  Xi,  X2 .  .  .  Xi._i,  Xi.^2i  ^^^  so  on. 

It  remains  to  be  shown  that  x^,  X2,  .  .  .  x^._-i  can  bo  taken  arbitra- 
rily, that  is,  that  the  assumption  H  =  H^  is  always  allowable.  Sup- 
pose that  i?i  contains  ccj , a?2 ? •  •  •  ^at- a •  Then  in  the  series  Hi,  H2,... 
there  is  a  group -Ho  which  also  contains  i?/,_a  +  i.  Proceeding  from 
if 0  and  the  elements  Xi,X2,.  .  .  x^,_a  +  i,  wo  construct  a  series  of 
groups,  as  before,  arriving  finally  at  the  group  H. 

§  83.  Tlieorem  XX.  If  a  prijnitive  group  G  of  degree  n 
contains  a  primitive  subgroup  H  of  degree  k  then  G  is  at  least 
(n — k-\-  l)-fold  transitive. 

From  the  preceding  theorem  Hi  affects  the  elements  x^x^,  .  .  .  x^-, 
\Hi,  H2\  the  elements  Xi,  X2,...x^.,x^..^i;  j if j ,  iJg ,  i^g }  the  ele- 
ments  Xi,  X.J,  .  .  ..Xf,,X/,.^i,  x^.^2'■,    and  so  on.     All  these  groups  are 


GENERAL    CLASSIFICATION    OF    GROUPS.  93 

transitive;  consequently,  from  Theorem  XIII,   {H^,  Hz}  is  two  fold 
transitive,  [H^^H^^H^]  three-fold  transitive,  and  finally 


r^\H,,...H, 


+  11 


is  at  least  (n  —  A;-|-l)-fold  transitive.     Therefore  G,  which  includes 
-T,  is  also  at  least  (n  —  A;  +  l)-fold  transitive.  * 

Corollary  I.  If  a  primitive  group  of  degree  n  contains  a 
circular  substitution  of  the  prime  order  p,  the  group  is  at  least 
{n — p  -\-  V)-fold  transitive. 

For  the  powers  of  the  circular  substitution  form  a  group  H  of 
degree  p. 

Corollary  II.     If  a  transitive  group  of  degree  n  contains  a 

2n 
circular  substitution  of  prime  order  p  <  -^,  then,    if    the  group 

does  not  contain  the  alternating  group,  it  is  non-primitive. 

From  Theorem  XV,  every  group  which  is  more  than  1-^  +  11- 

fold  transitive  is  either  alternating  or  symmetric.     And  since  the 

presence  of  a  circular  substitution  of  a  prime  order  p  in  a  primitive 

group  would  require  the  latter  to  be  at  least  (n — p  -f-  l)-fold  tran- 

2n 
sitive,  it  would  follow,  if  p  <  -^,  that  the  group  would  be  more  than 

o 

(  ^  + 1  I -fold  transitive  and  must  therefore  be  either  alternating  or 

symmetric.     As  these  alternatives  are  excluded,  the  group  must  be 
non-primitive. 

§  84  In  the  proof  of  Theorem  XIX  the  primitivity  of  the 
group  H  was  only  employed  to  demonstrate  the  presence  of  substi- 
tutions which  contained  two  successions  of  elements  of  a  certain 
kind.  The  presence  of  such  substitutions  would  also  evidently  be 
assured  if  H  were  two-fold  or  many-fold  transitive.  Theorems  XIX 
and  XX  would  therefore  still  be  valid  in  this  case.  The  latter  then 
takes  the  form: 

Theorem  XXI.     If  a  primitive  group  G  of  degree  n  con- 

♦Another  proof  of  this  theorem  is  given  by  Rudio:  Ueber  primitive  Gruppen, 
Crelle  CII,  p.  1. 


94 


THEORY    OF    SUBSTITOTIONS. 


tains  a  k-fold  transitive  subgroup  {k ^2)  of  degree  q,  then  G  is  at 
least  (n  —  q-\-  2)  transitive. 

§  85.  If  the  requirement  that  the  subgroup  H  of  the  preceding 
Section  shall  be  primitive  or  multiply  transitive  is  not  fulfilled,  the 
the  theory  becomes  at  once  far  more  complicated.  *  We  give  here 
only  a  few  of  the  simpler  results. 

Theorem  XXIT.  If  a  primitive  group  G  of  degree  n  con- 
tains a  subgroup  H  of  degree  A  <  n,  then  G  also  contains  a  subgroup 
whose  degree  is  exactly  n  —  1;  or  in  other  words:  A  transitive  group 
G  of  n  elements,  which  has  no  subgroup  of  exactly  n  —  1  elements^ 
but  has  a  subgroup  of  loiver  degree,  is  non-primitive. 

Suppose  that  the  subgroup  H  of  degree  >i  <  w  affects  the  ele- 

n 
ments  x^,X2,  .  ...  X},.     In  the  first  pl^ce  if  A  <  -^then  the  group  G, 

on  account  of  its  primitivity,  contains  a  substitution  s,  which  replaces 

one  element  of  x^,x<^, .  .  .  x^hj  another  element  of  the  same  system 

and  at  the  same  time  replaces  a  second  element  of  a?i ,  a?2  5  •  •  •  ^a  by 

some  new  element.     Then  H'  =  s{~  ^Hs^  contains  beside  some  of  the 

old  elements,  also  certain  new  ones,  so  that  iiTi  =  \H,H'\   affects 

more  than  X  elements,  but  less  than  n,  since  H  and  H'  together 

n 
affect  at  the  most  (2  A —  1)  <n.     If  the  degree  Aj  of  H^  is  still  <  -^, 

n 
we  repeat  the  same  process,  until  Aj  is  equal  to  or  greater  than  — . 

Suppose  that  the  elements  of  the  last  H^  are  x^,X2,  .  .  .  x^^.  Then 
the  primitive  group  G  must  again  contain  a  substitution  Sg  which 
replaces  two  elements  not  belonging  to  H^  by  two  elements,  one  of 
which  does,  while  the  other  does  not,  belong  to  H^ .  Then  the  group 
H\  =  S2H^S2~^  will  connect  new  elements  with  those  of  H^]  but, 
from  the  way  in  which  S2  was  taken,  one  new  element  is  still  not  con- 
tained in  H\ .  That  some  of  the  old  elements  actually  occur  in  H\ 
follows  from  the  fact  that  Aj  >  J  ?2.  Accordingly  if 2  =  i  ^1 ,  H'l  \  con- 
tains more  elements  than  H^  but  less  than  G.  Proceeding  in  this 
way,  we  must  finally  arrive  at  a  group  K  which  contains  exactly 
(n — 1)  elements. 

*C.  Jordan:  Liouville  Jour.  (2)  XVI.  B.  Marggraf:  Ueber  primitive  Gruppen  mit 
transitiven  Unlergruppen  geriugeren  Grades;  Giessen  Dissertation,  1890. 


GENERAL  CLASSIFICATION  OF  GROUPS.  95 

If  H  is  transitive,  then  H',  and  consequently  Hi=\H',  H\,  and 
80  on  to  Kj  are  also  transitive.  From  Theorem  XIII,  G  must 
therefore  in  this  case  be  at  least  two -fold  transitive.  We  have 
then  the  following 

Corollary.  If  a  primitive  group  G  contains  a  transitive 
subgroup  of  lower  degree,  then  G  is  at  least  two-fold  transitive. 

§  86.  We  turn  now  to  a  series  of  properties  based  on  the  the- 
ory of  self- conjugate  subgroups. 

Let  jff  =  [1,  S2,S3,  .  .  .  sj  be  a  self- conjugate  subgroup  of  a 
group  G  of  order  n  =  km.  The  substitutions  of  G  can  be  arranged 
(§  41)  in  a  table,  the  first  line  of  which  contains  the  substitutions 
of  H. 

*1  -—  Ij      ^2  J  ^3  ,  .  .  .  Stnj 

*^21  ^2^2i     ^3^2)      •  •  •  ^m^2i 

^3)  ^2^Sj      ^3*^3)     •  •   •  ^m'^Sj 


From  the  definition  of  a  self- conjugate  subgroup  we  have  then 

that  is,  the  line  of  the  table  in  which  the  product  {s^^^a)  (^/x'^'/s)  occurs 
depends  only  on  ^a  and  <y^ ,  or  in  other  words,  if  every  substitution 
of  the  oP^  line  is  multiplied  into  every  substitution  of  the  [i^^  line, 
the  resulting  products  all  belong  to  one  and  the  same  line. 

If  we  denote  the  several  lines,  regarded  as  units,  hj  z^,Z2,  .  .  .z^j 
then  the  line  containing  the  product  of  the  substitutions  of  Za  into 
those  of  z^  may  be  denoted  by  z^z^.  This  symbol  has  then  a  defi- 
nite, unambiguous  meaning.  Moreover,  ZaZ^  cannot  be  equal  to 
ZaZy  or  to  ZyZ^.  For  then  we  should  have  from  the  last  paragraph 
<^a'^/3  =  'yaS  ^^  '^a^/3  =  ^^v'^jS ,  that  is,  a^  —  ff^oT  (T^  —  ff^.  Conse- 
quently 


__  1^1 5         ^2)  '  •   '  Zij       •   •   •     I 

V^l  ^a^^i^ai  •   •  •  Z^Zg.  .  .  .  J 


denotes  a  substitution  among  the  2;'s,  and  this  substitution  corres- 
ponds to  all  the  substitutions  S},(^a  (>^  =  1,  2,  .  .  .  m)  of  the  a*^  line 
of  the  table.     The  fs  therefore  form  a  group  T,  which  is  (1-m)- 


96  THEORY    OF    SUBSTITUTIONS. 

fold  isomorphic  with  the  given  group  G.  The  degree  and  order  of 
T  are  equal,  and  both  are  equal  to  k  (Cf.  §  97).  To  the  identical 
substitution  of  T  corresponds  the  self-conjugate  subgroup  H  of  G. 

We  shall  designate  T  as  the  quotient  of  G  and  H ,  and  write 
accordingly  T—G\ H. 

§  87.  A  group  G  which  contains  a  self-conjugate  subgroup  H, 
different  from  identity,  is  called  a  compound  group;  otherwise  G  is 
a  simple  group.  If  G  contains  no  other  self-conjugate  subgroup  K 
which  includes  H,  then  i?is  a  maximal  self -conjugate  subgroup. 

If  G^  is  a  compound  group,  and  if  the  series  of  groups 
G,  Gj,  G^2j   • .  •    G^-,  1 

is  so  taken  that  every  G\  is  a  maximal  self- conjugate  subgroup  of 
the  preceding  one,  then  this  series  is  called  the  series  belonging  to 
the  compound  group  G,  or  the  series  of  composition  of  G,  or, 
still  more  briefly,  the  series  of  G. 

If  the  numbers 

r,   rj  =  r :  e^ ,   ^2  — ^  ^i  •  ^2  ?    •  •  •    ^/u.  ^^  ''ha  —  1  •  ^/u,  5   ^/u.  + 1  ^^  '"/*  •  ^/u.  + 1  =^  1 

are  the  orders  of  the  successive  groups  of  the  series  of  composition 
of  G,  then  ^i,  63?  •  •'•  ^/oi  +  i  are  called  the  factors  of  composition  of 
G]  and  we  have  r  =  e^  63 63 .  .  .  e^  + 1 . 

If,  in  accordance  with  the  notation  of  §  86,  we  write, 

G:Gi=^  r^,     Gj :  G^2  =  A  J      •  •  •      ^m  - 1  •  ^m  =  ^m  >     Gfi  •  1  ^=  A 

the  order  and  the  degree  of  every  1\  is  equal  to  Ca  («  =  1, 2, . . .  /x  +  1). 
All  the  groups  Fa  are  simple.  For  F^  is  (l-ra)-fold  isomorphic 
with  Ga-i,  and  to  the  identical  substitution  in  -/«  corresponds  Ga  in 
Ga-\.  Consequently,  if  !'„,  contains  a  self-conjugate  subgroup  dif- 
ferent from  identity,  then  the  corresponding  self- conjugate  sub- 
group of  Ga-i  (§  73)  contains  and  is  greater  than  Ga-  The  latter 
would  therefore  not  be  a  maximal  self- conjugate  subgroup  of  G^a-i. 

The  groups  /',  which  define  the  transition  from  every  Ga  to  the 
following  one  in  the  series  of  composition,  are  called  the  factor 
groups  of   (r.  * 

*  O.  Holder;  Math.  Ann.  XXXIV,  p.  30  ff. 


GENERAL    CLASSIFICATION    OF    GROUPS.  97 

§  88.  Given  a  compound  group  G,  it  is  quite  possible  that  the 
corresponding  series  of  composition  is  nob  fully  determinate.  It  is 
conceivable  that,  if  a  series  of  composition 

G,  Gi,  G2,  .  .  .  Gfj,,! 
has  been  found  to  exist,  there  may  also  be  a  second  series 

G,  G\,  G'2,  .  .  .  G'v,l 
in  which  every  G'  is  contained  as  a  maximal  self- conjugate  subgroup 
in  the  preceding  one.  We  shall  find  however  that,  in  whatever 
way  the  series  of  composition  may  be  chosen,  the  number  of  groups 
G  is  constant,  and  moreover  the  factors  of  composition  are  always 
the  same,  apart  from  their  order  of  succession. 

Suppose  the  substitutions  of  Gi  and  G\  to  be  denoted  by  Sa  and 
s'a  respectively.  Let  r^  =  r  :  e^  be  the  order  of  Gi ,  and  r\—r:  e\ 
that  of  G\ .  The  substitutions  common  to  Gi  and  G\  form  a  group 
r  (§  44),  the  order  x  of  which  is  a  factor  of  both  r^  and  r\ .  We 
write 

r^  =  xy,     r\  =  xy\ 

The  substitutions  of  F  we  denote  by  <?«•  All  the  substitutions  of  Gi 
may  then  be  arranged  in  a  table,  the  first  line  of  which  consists  of 
the  substitutions  t^  of  F.     We  obtain 

0-1  =  1,  (72)  ^S,    '   '   '        f^x\  A 

^2<^n       ^2'^2>      §2^3J    •   .   .    ^2^x1         §2^^J 


where  the  §  belonging  to  any  line  is  any  substitution  of  G^  not  con- 
tained in  the  preceding  lines.  The  group  G\  can  be  treated  in  the 
same  way.  We  will  suppose  that  in  this  case,  in  place  of  §i ,  §2 »  •  •  •  r 
we  have  s'l,  ^\^  .  .  .  Every  substitution  of  G'  or  G\  can,  then,  be 
written  in  the  form 

Again,  the  product 

belongs  to  G^.  For,  since  G'^G^G—G^,  it  follows  that  s'^-'s^^, 
which  occurs  in  the  second  form  of  the  product,  is  equal  to  s^?  and 
the  product  itself  is  equal  to  s'^s^.     But,  from  the  third  form,  this 


98  THEORY    OF    SUBSTITUTIONS. 

same  product  belongs  to  G',  since  G~^G'G=  G' ,  and  therefore 
Sa~^s'p'~^Sa  =  s'y,  SO  that  the  product  is  equal  to  s'ys'^.  Conse- 
quently the  product  belongs  to  the  group  F  which  is  common  to  Gi 
and  G\ .     Hence 

In  particular,  since  the  o-'s  belong  to  both  the  s's  and  the  s"s,  we 
obtain 

From  this  it  follows  that  the  substitutions  of  the  form  ^a^'^^y  form 
a  group  @ .  For,  by  repeated  applications  of  the  equation  B),  we 
obtain 

The  group  @  is  commutative  with  G;  for  we  have 

The  group  (S  is  more  extensive  than  Gi  or  G';  it  is  contained  in  G; 
consequently,  from  the  assumption  as  to  G^  and  G\  &  must  be 
identical  with  G. 

The  order  of  (S  is  equal  to  xy  ■  y'.  For,  if  §a§'^<^v  =  ^c^'i^c  it  is 
easily  seen  that  a  —  a,  b  =  /S,  c  =  y.  Consequently  the  order  of  G 
is  also  xyy\  and  since  we  have 

r  =  ViBi  =  xye^ ,     r  —  r\e\  =  xy'e\ 
it  follows  that 

y'=ei,     y  =  e\. 

r         7\       r\ 


This  last  result  gives  us  for  the  order  of  T,  x  = 


1    1         1       ^1 

We  can  show,  further,  that  T  is  a  maximal  self- conjugate  subgroup 
of  (tj  and  of  G\ ,  and  consequently  occurs  in  one  of  the  series  of 
composition  of  either  of  these  groups.  For  in  the  first  place  I\  as 
a  part  of  G\ ,  is  commutative  with  G^ ,  and,  as  a  part  of  G^ ,  is  com- 
mutative with  (r'l ,  so  that  we  have 

Gr  TG,  =  G\ ,    G'r  'rG\  =  g,  . 

But  since  the  left  member  of  the  first  equation  belongs  entirely  to 
Gi ,  the  same  is  true  for  the  right  member,  and  a  similar  result  holds 
for  the  second  equation.     Consequently 


GENEEAL    CLASSIFICATION    OF    GEOUPS.  99 

Gr'vG,  =  r,    G'r'rG\  =  r. 

Again  there  is  no  self-conjugate  subgroup  of  G^  intermediate  be- 
tween Gi  and  i'  which  contains  the  latter.  For  if  there  were  such  a 
group  H  with  substitutions  ta ,  then  it  would  follow  from  A)  that 

that  is,  iJ  is  also  commutative  with  G'l.  And  since  Gy  and  G'l 
together  generate  G,  it  appears  that  H  must  be  commutative  with 
G.  If  now  we  add  to  the  fa's  the  §'3?  S'3,  .  .  . ,  then  the  substitu- 
tions §'atp  form  a  group.  For  since  /'  is  contained  in  if  and  in  G^ 
we  have  from  A) 

This  group  is  commutative  with  G,  since  this  is  true  of  its  compo- 
nent groups  H  and  G\.  It  contains  G\^  which  consists  of  the  sub- 
stitutions S'a'^/s.  It  is  contained  in  G,  which  consists  of  the  substitu- 
tions §'a§i3S-  ^^^  this  is  contrary  to  the  assumption  that  G\  is  a 
maximal  self-conjugate  subgroup  of  G.  We  have  therefore  the  fol- 
lowing preliminary  result: 

If  in  two  series  of  composition  of  the  group  G,  the  groups  next 
succeeding  G  are  respeetively  G^  and  G\ ,  then  in  both  series  we  may 
take  for  the  group  next  succeeding  G^  or  G\  one  and  the  same  max- 
imal self-conjugate  subgroup  l\  which  is  composed  of  all  the  substi- 
tutions common  to  G^  and  G\.  If  e^  and  e\  are  the  factors  of 
composition  belonging  to  Gi  and  G\  respectively,  then  F  has  for  its 
factors  of  composition,  in  the  first  series  e\,  in  the  second  e^. 

§  89,     We  can  now  easily  obtain  the  final  result. 
Let  one  series  of  composition  for  G  be 

1)  G,  Gi,G2,  G3,  . . . , 

r,    r,  =  r:ei,  r.  —  r^-.e^,  r-^^r^-.e^,  .  .  .  , 
and  let  a  second  series  be 

2)  G,  G\,  G'2,  G's,  .  . , ,' 


r  '.e  I,  r  2  —  '^^1*^25  ^'3  —  ^'2*^3 


Then  from  the  result  just  obtained,   we  can  construct  two  more 
series  belonging  to  G: 


100  THEOEY    OF    SUBSTITUTIOXS. 

3)  G,  G„I\  J,H,...  4)  G,  G\,  F,  ^  H,  .  ,  . , 

r,r,  =  r:e,,  r^  —  r.-.e'^,..,,        r,r\  =  r:e\,  r'^  =  r\\e,  ..., 

and  apply  the  same  proof  for  the  constancy  of  the  factors  of  com- 
position to  the  series  1)  and  3),  and  again  2)  and  4),  as  was  employed 
above  in  the  case  of  the  series  1)  and  2).  The  series  3)  and  4) 
have  obviously  the  same  factors  of  composition. 

The  problem  is  now  reduced,  for  while  the  series  1)  and  2)  agree 
only  in  their  first  terms,  the  series  1)  and  3),  and  again  2)  and  4), 
agree  to  two  terms  each.  The  proof  can  then  be  carried  another 
step  by  constructing  from  1)  and  2)  as  before  two  new  series,  both 
of  which  now  begin  with  G,  G^ : 

1')         G,  G,,G^„@,$,3,..., 

J      'n      '2  —  'l'*^2  5     '      3  —  /2-t>2>'-' 

3')       G,G„  r,®,o,3,..., 

^>      ^]  5     ^  2  =^  '^1  •  ^  2  ?     ^     3  ^  '^  2  •  ^2  )   •   •   • 

These  series  have  again  the  same  factors  of  composition,  and  1')  and 
1)  and  again  3')  and  3)  agree  to  three  terms,  and  so  on. 
We  have  then  finally 

Theorem  XXIII.  //  a  compound  group  G  admits  of  two 
different  series  of  composition,  the  factors  of  composition  in  the  two 
cases  are  identical,  apart  from  their  order,  and  the  number  of 
groups  in  the  two  series  is  therefore  the  same. 

§  90.     From   §  88  we  deduce   another   result.     Since    G~^rG 

belongs   to   Gi,  because    G~'^GiG=  G^,    and  also  to  G\   because 

G~^G'iG=  G\,  it  appears  that  G~^rG,  as  a  common  subgroup  of 

Gi  and  G\,  must  be  identical  with  T,  so  that  F  is  a  self- conjugate 

subgroup  of  G.     From  §  86  it  follows  that  it  is  possible  to  con- 

r 
struct  a  group  ii  of  order  e^e'^  which  is  (1 ^)-fold   isomorphic 

with  G,  in  such  a  way  that  the  same  substitution  of  i2  corresponds 
to  all  the  substitutions  of  G  which  only  difPer  in  a  factor  (t.  We  will 
take  now,  to  correspond  to  the  substitutions  1,  §0?  ^35  •  •  •  ^/i  ot  Gi, 
the  substitutions  1, 10^,0)^,  .  .  .  w/^  of  i2,  and,  to  correspond  to  the 
1,  §'2,  §'3,  .  .  .  S'^j  of  G'l,  the  substitutions  1,  (o'^,  w'3,  .  .  .  <y',,  of  P-.  In 
no  case  is  §'a  =  §|3§-y,  for  the  <r's  form  the  common  subgroup  of  Gi 


GENERAL    CLASSIFICATION    OF    GROUPS.  101 

and  G'l.     Consequently  the  w's  are  different  from  the  w"s.     Both 
classes  of  substitutions  give  rise  to  groups: 

^1  =  [1,  ^2,  ^3,  .  •  .  ^e\],        ^2  =  [1,  ^'2,  ^'3,   .  .  .  ">M, 

and,  since  Sa^'is  =  ^'^^a^^-y,  it  follows  that  ^^  ^\  =  ^\  P^^ .     Moreover 
every  s  in  G  is  equal  to  Sa^'^S,  that  is  ^  =  P]P\. 

We  obtain  P  therefore,  by  multiplying  every  substitution  of  P^ 
by  every  one  of  P\. 

§  91.  We  consider  now  two  successive  groups  of  a  series  of 
composition,  or,  what  is  the  same  thing,  a  group  G  and  one  of  its 
maximal  self -conjugate  subgroups  H.  Suppose  that  s\  is  a  substi- 
tution of  G  which  does  not  occur  in  H,  and  let  s\"'  be  the  lowest 
power  of  s\  which  does  occur  in  H  {m  is  either  the  order  of  s\  or 
a  factor  of  the  order).  If  m  is  a  composite  number  and  equal  to 
p  g,  we  put  s V  =  ^1 )  and  obtain  thus  a  substitution  Sj  which  does  not 
occur  in  IT,  and  of  which  a  prime  power  s^^  is  the  first  to  occur  in  H. 
We  then  transform  Sj  with  respect  to  all  the  substitutions  of  G,  and 
obtain  in  this  way  a  series  of  substitutions  Si,  S2,  .  .  .  s^.  No  one  of 
these  can  occur  in  H.  For  if  this  were  the  case  with  Sa  =  c>~^Si(t, 
then  <7Sa^~^  =  s^,  being  the  transformed  of  a  substitution  Sa  of  H 
with  respect  to  a  substitution  (r~^  oi  G,  would  also  occur  in  H. 

We  consider  then  the  group 

r={H,s„S2,...s^\. 

This  group  contains  H  and  is  contained  in  G.  If  t  is  any  arbitrary 
substitution  of  G,  we  have 

t-Tt  =  t-'{Hsi''s.f.  ..\t  =  t-'Ht-t-'s'^t .  t-'s/t .  .  . 

=  Hs,^%.^...  =r. 

r  is  therefore  commutative  with  G.  These  three  properties  of  F 
are  inconsistent  with  the  assumption  that  H  is  a  maximal  self-con- 
jugate subgroup  of  G,  unless  F  and  H  are  identical. 

If  we  remember  further  that  all  substitutions,  as  Si,S2,  .  .  .  s^, 
which  are  obtained  from  one  another  by  transformation,  are  similar, 
we  have 

Theorem  XXIV.  Every  group  of  the  series  of  composition 
of  any  group  G,  is  obtainable  from  the  next  following  {or,  every 
group  is  obtainable  from  any  one  of  its  maximal  self- conjugate 


102  THEORY    OF    SUBSTITUTIONH. 

subgroups)  by  the  addition  of  a  series  of  substitutions^  1)  ivhich  are 
similar  to  one  another,  and  2)  a  prime  poicer  of  ivhich  belongs  to 
the  smaller  group.  The  last  actual  group  of  a  series  of  composi- 
tion consists  entirely  of  similar  substitutions  of  prime  order. 

§  92.  The  following  theorem  is  of  great  importance  for  the 
theory  of  equations: 

Iriieorem  XXV,  The  series  of  composition  of  the  symmet- 
ric group  of  n  elements,  consists,  if  n^  4,,  of  the  alternating  group 
and  the  identical  substitution.  The  corresponding  factors  of  com- 
position are  therefore  2  and  ^n\  The  alternating  group  of  more 
than  four  elements  is  simple. 

We  have  already  seen  that  the  alternating  group  is  a  maximal 
self- conjugate  subgroup  of  the  symmetric  group.  It  only  remains 
to  be  shown  that,  for  n  >  4,  the  alternating  group  is  simple.  The 
proof  is  perfectly  analogous  to  that  of  §  52,  and  the  theorem  there 
obtained,  when  expressed  in  the  nomenclature  of  the  present  Chap- 
ter, becomes:  a  group  which  is  commutative  with  the  symmetric 
group  is,  for  n  >  4,  either  the  alternating  group  or  the  identical  sub- 
stitution. It  will  be  necessary  therefore  to  give  only  a  brief  sketch 
of  the  proof. 

Suppose  that  H^  is  a  maximal  self -con  jugate  subgroup  of  the 
alternating  group  H,  and  consider  the  substitutions  of  H^  which  afPect 
the  smallest  number  of  elements.  All  the  cycles  of  any  one  of  these 
substitutions  must  contain  the  same  number  of  elements  (§  52). 
The  substitutions  cannot  contain  more  than  three  elements  in  any 
cycle.     For  if  H  contains  the  substitution 

s  —  ypc-YJC^uC^iC  ^ ...).,., 
and  if  we  transform  s  with  respect  to  ^r  =  {x2X.iX^,  which  of  course 
occurs  in  H,  then  s~  V~^srt-  contains  fewer  elements  than  s. 

Again  the  substitutions  of  H^  with  the  least  number  of  the  ele- 
ments cannot  contain  more  than  one  cycle.     For  if  either 

Sa  ^=  (^^1^2)  y^a'^*)  .  .  .  J  Sj3  =  {X^X2Xj)  {X^X^X^j  .  .  .  , 

occurs  in  H,  and  if  we  transform  with  respect  to  <t  =  {XiX2Xz),  the 
products 

will  contain  fewer  elements  than  Sa,  sp  respectively. 


GENERAL    CLASSIFICATION    OF    GROUPS.  103 

The  substitutions  which  affect  the  smallest  number  of  elements 
are  therefore  of  one  or  the  other  of  the  forms 

The  first  case  is  impossible,  since  the  alternating  group  cannot  con- 
tain a  transformation.  The  second  case  leads  to  the  alternating 
group  itself. 

If  n  =  4,  we  obtain  the  following  series  of  composition :  1)  the 
symmetric  group;  2)  the  alternating  group;  3)  (t2  =  [1,  (iCjOJa)  (x^x^), 
(x^x^)  (X2X,),  {x,x^)  {x.2Xs)'];  3)  Gs  =  [1,  (0^1X2)  (x^x^)']]  5)  G^  =  1.  The 
exceptional  group  G2  is  already  familiar  to  us. 

§  93.     We  may  add  here  the  following  theorems : 

Theorem  XXVI.  Every  group  G,  which  is  not  contained 
in  the  alternating  group  is  compound.  One  of  its  factors  of  com- 
position is  2.     The  corresponding  factor  group  is   [(1,  2^1 ,  2:2,)]. 

The  proof  is  based  on  §  35,  Theorem  VIII.  The  substitutions 
of  G  which  belong  to  the  alternating  group  form  the  first  self-con- 
jugate subgroup  of  G. 

Theorem  XXVII.  //  a  group  G  is  of  order  p%  p  being  a 
prime  number,  the  factors  of  composition  of  G  are  all  equal  to  p. 

The  group  K  of  order  p^  obtained  in  §  30  is  obviously,  from  the 
method  of  its  construction,  compound.  It  contains  a  self- conjugate 
subgroup  L  of  order  p^~^  and  this  again  contains  a  self- conjugate 
subgroup  M  of  order  p^~^,  and  so  on.  The  series  of  composition 
of  K  consists  therefore  of  the  groups 


K,     L,     M,     . 

..  Q,   R,  . 

..S,l, 

of  orders 

p',p'-\p'-\  . 

..p^p^-',. 

..p,i. 

The  last  corollary  of  §  49  shows  that  we  need  prove  the  present  theo- 
rem only  for  the  subgroups  of  K.  If  G  occurs  among  these  and  is 
one  of  the  series  above,  the  proof  is  already  complete.  If  G  does 
not  occur  in  this  series,  suppose  that  R  is  the  first  group  of  the 
series  which  does  not  contain  G,  while  G  is  a.  subgroup  of  Q.  We 
apply  then  to  G  the  second  proposition  of  §  71.  Suppose  that  H  is 
the  common  subgroup  of  R  and  G.  Then  IZ"  is  a  self-conjugate 
subgroup  of  G,  and  its  order  is  a  multiple  of  p°-~^  and  is  conse- 


104 


THEORY    OF    SUBSTITUTIONS. 


quently  either  p**"^  or  p".     The  latter  case  is  impossible  since  then 
.  G  would  be  contained  in  R.     Consequently  H  is  of  order  p*~\  and 
the  theorem  is  proved. 

Theorem  XXVIII.     //  a  group   G  of  order  r  contains   a 

r 
self-conjugate  subgroup   H  of  order  —  then  no  substitution  of  G^ 

which  does  not  occur  in  H  can  be  of  an  order  prime  to  e.  * 

We  construct  the  factor  group  r=  G:H  oi  the  order  e.  No  one 
of  the  substitutions  of  F,  except  the  identical  substitution,  is  of  an 
order  prime  to  e.  To  any  substitution  s  of  G  which  does  not  occur 
in  H  corresponds  a  (t  which  is  different  from  1.  On  account  of  the 
isomorphism  of  G  and  F,  there  corresponds  to  every  power  s^'  of  s 
the  same  power  </'  of  <r.  If  /.  is  the  lowest  power  of  s  for  which 
s**  =  1,  "then  at  the  same  time  o-"  =  1.  z  is  to  therefore  a  multiple  of 
the  order  of  (t  and  consequently  is  not  prime  bo  e. 

If,  in  particular,  e  is  a  prime  number,  then  the  order  of  every 
substitution  of  G  which  is  not  contained  in  H  is  divisible  by  e. 

§  94.  Among  the  various  series  of  composition  of  a  group  G, 
the  principal  series  of  composition,  or  briefly,  the  principal  series, 
is  of  special  importance  in  the  algebraic  solution  of  equations.  This 
principal  series  is  obtained  from  any  series  of  composition  by  re- 
taining only  those  groups  of  the  series  which  are  themselves  self- 
conjugate  subgroups  of  G.     Suppose  the  resulting  series  to  be 

G,H,J,...  K,  L,  M,  1. 

The  series  of  G  may  itself  be  the  principal  series.  This  will  be 
the  case,  for  example,  as  we  shall  immediately  show,  if  all  the  fac- 
tors of  composition  of  the  series  are  prime  numbers. 

Assuming  that  the  principal  series  is  not  identical  with  the  given 
series,  suppose  that  the  latter  contains,  for  instance  between  H  and 
J,  other  groups,  as 

H^,  H2, .  .  .  Hy^l, 

Hi  is  therefore  commutative  with  H,  but  not  with  G.     Consequently 
II~  H-yH  '=■  Hi ,     G~  Hi  G=j=Hi . 

*  A.  Kneser:  Ueber  die  algebraische  Unauflosbarkeit  hoherer  Gleicliungen.  Crelle 
CVI.,  pp.  59-60. 


GENEEAL    CLASSIFICATION    OF    GEOUPS. 


105 


Accordingly,  if  we  transform  -ffj  with  respect  to  all  the  substitutions 
of  G,  we  shall  obtain  a  series  of  groups  H^ ,  H\ ,  H'\ ,  .  .  .  All  of 
these  are  contained  as  self- conjugate  subgroups  in  iT,  for  if  a  is  any 
substitution  of  G,  then  (t~^Hi(t  =  H\  is  contained  in  <t~^H(t  =  H. 
Moreover 

for  if  T  is  any  substitution  of  M,  then  from  <7~V<r  =  v  follows 
<.-V-V  =  v-'(c/.§36). 

Again  J  is  contained  in  every  one  of  the  groups  H^,H\ ,  H'\,. . . 
For  J  is  contained  in  H^ ,  and  consequently  <r  ~  ^Jff  =  J  is  contained 
inff-^  H^(T  =  H\,  and  so  on.  Finally  H\ ,  like  Hi/is  a  maximal  self- 
conjugate  subgroup  of  H.  For  if  there  were  any  self- conjugate  sub- 
group between  H  and  H\ ,  then  the  same  would  be  true  of  H  and 
H^ .  In  fact  if  H\  is  obtained  from  if i  by  transformation  with 
respect  to  <x,  then  the  intermediate  group  between  H  and  H^  would 
proceed  from  that  between  H  and  H\,  by  transformation  with 
respect  to  (t~^.  In  the  series  of  G,  the  group  H  may  therefore  be 
any  one  of  the  several  groups  of  the  same  type  Hi,H\,  .  .  .  All 
of  these  belong  to  the  same  factor  e  of  composition,  £  being  the 
quotient  of  the  orders  of  H  and  H^ .  In  accordance  with  the  pre- 
liminary result  of  §  88,  we  can  then  continue  the  series  of  G  by 
taking  for  the  group  next  succeeding  H^  the  substitutions  common 
to  H^  and  H\,  or  to  H^  and  H'\,  or  to  H^  and  H"\,  and  so  on. 
From  the  same  result  the  new  groups  all  belong  to  the  same  factor 
of  composition  e.  Every  one  of  them  contains  J.  We  need  of 
course  consider  only  the  different  groups  among  them.  If  there  is 
only  one,  this  must  coincide  with  J.    For  the  entire  system  of  groups 

■til,  ^llJ^)J^       !»•••) 

and  consequently  the  group  common  to  all  of  them,  is  unaltered  by 
transformation  with  respect  to  G.  The  order  of  J  is  therefore 
obtained  by  dividing  that  of  H  by  el 

But  if  there  are  several  different  groups,  we  can  then  proceed 
in  the  same  way.  The  substitutions  common  to  H^,  H\,  H'\,  for 
example,  form  a  group  which  in  the  series  of  G  can  succeed  the 


106  THEORY    OF    SUBSTITUTIONS. 

^roup  composed  of  the  substitutions  common  to  H^  and  H\ .  The 
corresponding  factor  of  composition  is  again  e. 

After  V  repetitions  of  this  process  we  arrive  at  the  group  J.  The 
order  of  J  is  therefore  obtained  by  dividing  that  of  H  by  e".  The 
last  system  before  J  consists  of  >  groups  Hy_^,  H'v-i  .  .  .  which  are 
all  similar  and  all  belong  to  the  factor  e,  and  which  give 

Theorem  XXIX,  If  a  series  of  composition  of  G  does 
not  coincide  with  a  principal  series,  hut  if,  betiveen  tivo  groups  H 
and  J  of  the  latter,  v  —  1  groups  Hi^H^,  .  .  .  H^_i  of  the  former  are 
inserted,  theyi  to  H^,  Hoy  ...  J  belong  the  same  factoids  of  composition 
e,  and  the  order  r'  of  G  is  therefore  equal  to  the  order  r"  of  J  mul- 
tiplied by  £*'.  H  can  be  obtained  from  J  by  combining  with  J  a 
series  of  v  groups  Hy_^,  H\_^,  .  .  .  ,  ivhich  are  all  similar,  and  of 
the  order  r"s. 

Corollary  I.  If  the  factors  of  composition  of  a  group  are 
not  all  equal,  the  group  has  a  principal  series. 

Corollary  II,  Every  non-primitive  group  is  compound  if 
it  contains  any  substitution  except  identity  which  leaves  the  several 
systems  of  non-primitivity  unchanged  as  units.  If  the  group  con- 
tains greater  (including)  and  lesser  (included)  systems  of  non-prim- 
itivity, it  has  a  principal  series. 

The  instance  of  the  group 

G  =  [1,  (X^X^)  (x^Xi)  (x^Xq),  (x^X^)  (X2X^)  (XiXe),  (xiX^)  (x^x^)  (x^x^), 
[XiX^X^)  [X2X^Xq),  yXiXr^X^)  {^X2XqX2) J 

shows  that  non-primitivity  may  occur  in  a  simple  group.  In  this 
case  the  only  substitution  which  leaves  the  systems  Xi,X2i  X3,x^, 
and  x^,  a?6  unchanged  is  the  identical  substitution. 

Corollary  III,  The  groups  H^_^,H\_^,H'\_^,  .  . .  are 
commutative,  i.  e.,  the  equations  hold 


^,_/«)  H,_,^^)  =  H^-i^^^  H, 


(*). 


For    in   the    series    preceding   J  we   may  assume   the   sequence 
iZ"v_iH  ji2'^_iH-H'v-/^^h  •  •  •  to  occur.     Accordingly  we  must  have 


GENEEAL    CLASSIFICATION    OF    GROUPS.  107 

or 

Corollary  IV.  The  last  actual  group  M  of  the  principal 
series  of  G  is  composed  of  one  or  more  groups  similar  to  one  another, 
which  have  no  substitutions  except  identity  in  common,  and  which 
are  commutative  with  one  another. 

§  95.  We  have  now  to  consider  the  important  special  case 
where  £  is  a  prime  number  p. 

Instead  of  if'v_i,  -ff"v_i,  ...  we  employ  now  the  more  conven- 
ient notation 

H\  H'\  H"\  .  .  .  ifW. 

Then  H'  is  obtained  from  J  by  adding  to  the  latter  a  substitu- 
tion ^1 ,  the  p^^  power  of  which  is  the  first  to  occur  in  J.  We  may 
write  (§  91) 

H'  =  t,^J,    H^'^t^^'J,    ir'"=f3«J,...    (a  =  0, 1,  .  ..p— 1). 

Since  J  is  a  self- conjugate  subgroup  of  every  one  of  the  groups 
H',H'\  ...,wehave 

t,-''jti''  =  j,  t^-'^jt.'^^j,  tr''Jtz-  =  J, '■  ' 

and,  if  we  denote  the  substitutions  of  J  by  i^ ,  % ,  e's ,  .  .  . , 

t J         2.J         tj     2'2,  ^2         ^1         ^2     ^  ^3  5  ^3         *1         ^3     ~~  ^4 )   •   •   •  ? 

t\   h  ^  Hh     (*2%)i         ^2   H  ^-^  ^1^2'*  (%*!/?         *3   H  ^-  ^1^3     V*4*lj)   •   •   •   ) 

that  is,  the  substitutions  of  H\  of  H'\  of  H"\  and  so  on,  are  com- 
mutative among  themselves,  apart  from  a  factor  belonging  to  J. 

Since  we  can  return  from  J"  to  iJ  by  combining  the  substitutions 
of  H'  and  H'\  for  example,  into  a  single  group  (§  88),  we  have  from 
§  94,  Corollary  III 

^2        ^1^2  """  *^1    ^15         ^1        ^2        ^1  ^2    ^2) 

and  consequently,  by  combination  of  these  two  results, 

t\        ^2        ^1^2  ~~  t^   t'^2         ~~  ^2  % ) 

=  t^        ti°-l  1  =  f  1*        *i  5 
t\^        ^1  ^^  *2  ^3  • 

The  left  member  of  the  last  equation  is  a  substitution  of  H\  the 


108  THEORY    0¥    SUBSTITUTIONS.- 

right  member  a  substitution  of  H".  Since  these  two  groups  have 
only  the  substitutions  of  J  in  common,  the  powers  of  ^i  and  /,  must 
disappear.     Consequently  a  =  1,  /?= — 1,  and 

{tih)(i2h)={¥2){tih)is, 
(t,%)(t/i,)  =  {t/H)(t,%)i,. 

The  substitutions  of  the  group  formed  from  J,  t^,  and  1.2  are 
therefore  commutative  among  themselves,  apart  from  a  factor  be- 
longing to  J.  The  same  is  true  of  the  group  formed  from  J,  t^, 
and  fg,  or  from  J,  t.y,  and  ^3,  and  consequently  of  the  group 
\J,ti,t2,tr^\,  and  so  on,  to  the  group  H  itself.  (It  is  to  be  noted 
that  Corollary  III  of  §  94  involves  much  less  than  this.  There  it 
-was  a  question  of  the  commutativity  of  groups,  here  of  the  single 
substitutions.) 

Every  two  substitutions  of  H  are,  then,  commutative  apart  from 
a  factor  belonging  to  J.  We  will  prove  now  the  converse  proposi- 
tion: If  two  substitutions  of  H  are  commutative  apart  from  a  fac- 
tor belonging  to  J,  then  e  is  a  prime  number.  In  fact  this  will  be 
the  case,  if  the  substitutions  of  H'  have  this  property.  For,  this 
being  assumed,  if  £  were  a  composite  number,  suppose  its  prime  fac- 
tors to  be  g,  q\  q'\  .  .  .  We  select  from  H\ ,  in  accordance  with 
Theorem  XXIV,  §  91,  a  substitution  t  which  is  not  contained  in  J. 
The  lowest  power  of  t  which  occurs  in  J  will  then  be,  for  example, 
P.     Transforming,  we  have 

and,  since  by  assumption,  t'^H'  =  H'f^J, 

H'-\t''J)H'  =  t''J. 

The  group  \t,  J\  is  therefore  a  self-conjugate  subgroup  of  H\  which 
contains  J  and  is  larger  than  J.  Moreover,  it  is  contained  in  H', 
and  is  smaller  than  H'.  For,  if  t  is  commutative  with  J,  then  from 
§§  37-8  the  order  of  \t,J\  i8r'q<r''e.  This  is  contrary  to  the 
assumption  that  J  is  the  group  immediately  following  H'  in  the 
series  of  G. 


GENERAL    CLASSIFICATION    OF    GEOUPS.  109' 

Theorem  XXX.  //,  in  the  principal  series  of  composi- 
tion of  G,  the  order  r'  of  H  is  obtained  from  the  order  r"  of  J  by 
multiplication  by  p",  where  the  prime  number  p  is  the  factor  of  com- 
position for  the  intervening  groups  in  the  series  of  G,  then  the 
substitutions  of  H  are  commutative  among  themselves  apart  from 
factors  belonging  to  J.  Conversely,  if  this  is  the  case,  the  factors 
of  composition  of  the  groups  between  H  and  J  are  all  equal  to 
the  same  prime  number  p. 

§  96.  We  turn  finally  to  certain  properties  of  groups  in  rela- 
tion to  isomorphism.  .  . 

If  L  is  a  maximal  self -conjugate  subgroup  of  G,  and  .4  the 
corresponding  group  of  l\  then  A  is  also  a  maximal  self -conjugate 
subgroup  of  T. 

For  if  r  contained  a  self-conjugate  subgroup  S,  which  con- 
tained A ,  then  the  corresponding  group  T  oi  G  would  contain  L. 

The  series  of  composition  of  G  corresponds  to  that  of  F.  If 
G  and  F  are  simply  isomorphic,  all  the  factors  of  the  one  group  are 
equal  to  the  corresponding  factors  of  the  other.  But  if  G  is  mul- 
tiply isomorphic  to  F,  then  there  occur  in  the  series  for  G,  besides 
the  factors  of  F,  also  a  factor  belonging  to  the  group  S  which  cor- 
responds to  the  identical  substitution  of  F. 

The  proof  is  readily  found. 

If  G  is  multiply  isomorphic  to  F,  then  G  is  compound,  and  S  is  a 
group  of  the  series  of  composition  of  G. 

§  97.  Suppose  that  G  is  any  transitive  group  of  order  r,  affect- 
ing the  n  elements  x^,  X2,  .  .  .  x,,.  We  construct  any  arbitrary  n\- 
valued  function  c  of  x-^,X2,,  .  .  .  x,„  denote  its  different  values  by 
Ci ,  C2 ,  .  .  .  c„ , ,  and  apply  to  any  one  of  these,  as  Cj ,  all  the  substitu- 
tions of  G.     Let  the  values  obtained  from  Cj  in  this  way  be 

^15   ^2)  ?3?   •   •   •  ^v 

The  r  substitutions  of  G  will  not  change  this  system  of  functions  as 
a  whole,  but  will  merely  interchange  its  individual  members,  produ- 
cing r  rearrangements  of  these,  which  we  may  also  regard  as  sub- 
stitutions. These  substitutions  of  the  I's,  as  we  have  seen,  form 
a  new  group  F.      The  group  /'  is  transitive,  for  G  contains  substi- 


110  THEORY    OF    SUBSTITUTIONS. 

tutions  which  convert  ^^  into  any  one  of  the  values  I'l ,  ^2  >•••?,  > 
and  therefore  the  substitutions  of  F  replace  $1  by  any  element 
^n  ^2  J  •••'?»•.  Again  every  substitution  of  G  alters  the  order  of 
^1,  ^2)  •  •  .  ^r)  for  I  is  a  n! -valued  function.  Consequently  every  sub- 
stitution of  /'also  rearranges  the  Ci,?2)--*^r-  The  order  of  F  is 
therefore  equal  to  its  degree,  and  both  are  equal  to  r. 

G  and  F  are  simply  isomorphic.  For  to  every  substitution  of  G 
corresponds  one  substitution  of  F,  and  conversely  to  every  substitu- 
tion of  F  at  least  one  substitution  of  G.  And  in  the  latter  case  it 
can  be  only  one  substitution  of  G,  since  G  and  F  are  of  the  same 
order. 

Theorem  XXXI.  To  any  transitive  group  of  order  r  cor- 
responds a  simply  isomorphic  transitive  group,  the  degree  and  order 
of  which  are  both  equal  to  r.     Such  groups  are  called  regular. 

§  98.  Theorem  XXXII.  Every  substitution  of  a  regular 
group,  except  the  identical  substitution,  affects  all  the  elements.  A 
regular  group  contains  only  one  substitution  which  replaces  a  given 
element  by  a  prescribed  element.  Every  one  of  its  substitutions 
consists  of  cycles  of  the  same  order.  If  two  regular  groups  of  the 
same  degree  are  {necessarily  simply)  isomorphic,  they  are  similar 
i.  e.,  they  differ  only  in  respect  to  the  designation  of  the  elements. 
Every  regular  group  is  non-primitive.  * 

The  greater  part  oE  the  the  theorem  is  already  proved  in  the 
preceding  Section,  and  the  remainder  presents  no  difficulty.  We 
need  consider  in  particular  only  the  last  two  statements. 

Suppose  that  /',  with  elements  $1,  ?2)  •  •  •  '^n  and  substitutions 
<Ti ,  0-2 ,  . .  .  «T„  is  isomorphic  to  G  with  elements  ccj ,  a?2 ,  .  .  .  a?„  and 
substitutions  Si,  83,  .  .  .s,„  the  isomorphism  being  such  that  to  every 
<rx  corresponds  Sa-  Then  we  arrange  the  elements  x^  and  c^  in  pairs 
as  follows.  Any  two  of  them,  .Tj  and  Ci,  form  the  first  pair.  If 
then  Sx  converts  x^  into  x^^  and  if  the  corresponding  (j^  converts  Ci 
into  ca.,  then  Xk  and  ^x  form  a  second  pair.  No  inconsistency  can 
arise  in  this  way,  for  there  is  only  one  substitution  which  converts 
a?j  into  Xa,.  We  have  now  to  prove  that,  if  s^  contains  the  succes- 
sion XaXi,,  then  ^r^  contains  the  succession  $„  ^i,. 

*A  regular  group  of  prime  flegree  is  cyclical. 


GENERAL    CLASSIFICATION    OF    GROUPS.  Ill 

We  have 

5^  =  .  .  .  XiX„  •  •  •  )       Sj  =  .  .  .  X^Xi  ...  J       S»       Si)  —  ...  XaXf,  .  .  .  ) 
(T^—   .  .  .  $i^«  .  .  .  ,         (Tj,=:   .  ,  ,  $j*j  .  .  .  ,         (Ta~'ffb  =    •  .  •  I'al'fe  •   •   •  , 

and  since  there  is  only  one  substitution  which  replaces  Xa  by  a^^,  it 
follows  that 

Sa~^Si)  —  Sx,        <7a~^<^b  —  (^k-      ■ 

If  therefore  Sk  contains  a  cycle  composed  of  a  given  number  of  ele- 
ments Xa,  then  (Jk  contains  an  equal  cycle  composed  of  the  corres- 
ponding elements  c^.  Therefore  Sa  and  ff^,  and  consequently  G 
and  /'  are  of  the  same  type. 

The  last  part  of  the  theorem  is  proved  as  follows.  If  a  regular 
group  G  contains  a  substitution  s  =  {xxX2  .  .  .  x„)  {x^j^-^x^j^^-  ••)••• 
then  it  cannot  also  contain  t  =  (x^X2 .  .  .)  (a?4a?„,^2-  ..)...  For  we 
should  then  have  st'~'^  =  {xi)  (cc^  +  iX^ ...)...,  and  G  would  not  be  a 
regular  group.  Consequently  a?i,  Xg,  .  . .  x,„,  i.  e.,  the  elements  of  any 
arbitrary  cycle,  form  a  system  of  non-primitivity.  (The  remaining 
systems  however  are  not  necessarily  formed  from  the  remaining 
cycles  of  the  same  substitution). 

§  99.  If  the  groups  G  and  /'are  isomorphic,  and  if  G  is  intran- 
sitive, then,  if  in  every  substitution  of  G  we  suppress  all  elements 
which  are  not  transitively  connected  with  any  one  among  them,  as 
x^,  the  remaining  portions  of  the  several  substitutions  form  a  new 
transitive  group  G^  also  isomorphic  with  F.  It  may  however  hap- 
pen that  the  order  of  isomorphism  of  F  to  G^  is  increased.  Again, 
if  X2  is  any  new  element,  not  transitively  connected  with  x^ ,  we  can 
then  form  a  second  transitive  group  G2,  isomorphic  to  /'  and  con- 
taining X2,  and  so  on. 

The  intransitive  group  G  can  therefore  be  decomposed  into  a 
system  of  transitive  groups  isomorphic  with  T,  and  conversely  every 
intransitive  group  can  be  compounded  from  transitive  groups 
Gi,  G2,  .  .  .  In  the  case  of  simple  isomorphism  it  is  only  neces- 
sary to  multiply  the  several  constituents  directly  together. 

§  100.  Suppose  that  G  is  a  transitive  group  of  degree  m  and 
order  r  =  mmi,  which  is  fc-fold  isomorphic  with  a  second  group 
r.  Let  the  elements  of  G  he  x^^  X2,  .  .  .  x,^,  and  let  Gi  be  the  sub- 
group of  G  which  does  not  affect  x^ .     The  order  of  Gi  is  therefore 


112  THEORY    OF    SUBSTITUTIONS. 

mj .  If  82,83,  ,.  .  are  substitutions  of  G  which  convert  x-^  intO' 
X2,  x.^,  .  .  .  then  G^'S.2,  G^-s^,  . .  .  comprise  in  each  case  all  and  only 
the  substitutions  which  produce  the  same  effects. 

Suppose   that   to   G^  in   G  corresponds  I\  in  T,    the   order  of 
Tj  being  m^  k.     The  order  of  F  is  rk.     Consequently,  if  the  func- 

yk 
tion  (pi  belongs  to  F^ ,  then  <pi  takes  exactly  — -  —  m  values  under  the 

operation  of  all  the  substitutions  of  F.  Suppose  that  the  substitu- 
tion (7.2  of  F  which  corresponds  to  82  of  G  converts  ^1  into  ^2  •  Then 
F .  '62  contains  all  the  substitutions  which  convert  ^p^  into  ip2 .  Simi- 
lar considerations  hold  for  0-3  and  Cg,  a^  and  (p^,  and  so  on  to  g„^  and 
^^.     If  we  apply  all  the  substitutions  of  F  to  the  system  of  values 

we  obtain  rearrangements  which  can  be  regarded  as  substitutions  of 
the  new  elements  <p.  The  order  of  the  new  group  H  is  equal  to  the 
quotient  of  kr  by  the  number  of  substitutions  of  F  which  leave  all 
the  cp's  unchanged.  These  correspond  to  the  substitutions  of  G 
which  leave  all  the  a^'s  unchanged,  i.  e.,  to  the  identical  substitu- 
tion. To  this  correspond  k  substitutions  of  T,  and  consequently  the 
order  of  H  is  r. 

G  and  H  are,  then,  the  same  degree  m^ ,  of  the  same  order  r,  and 
they  are  isomorphic  and,  in  fact,  similar.  For  if  s  is  a  substitution 
of  G  which  replaces  Xq  by  Xa,  then  s  belongs  to  the  system  8a~^Gi8p. 
The  corresponding  substitution  of  H  is  obtained  by  applying  a  sub- 
stitution (T^^F^a^  to  the  system  f'l,  9^2?  •  •  •  ^m-  Every  one  of  these 
substitutions  replaces  9?^  by  (f^.  Accordingly  the  substitutions  of  H 
only  difPer  from  those  of  G  in  the  fact  that  the  latter  contain  a?'s 
where  the  former  contain  the  corresponding  ^''s. 

We  can  therefore  construct  all  groups  G  (or  H)  isomorphic  to  F 
by  applying  all  the  substitutions  of  F  to  any  function  (f  belonging  to 
any  arbitrary  subgroup  /\  of  /',  and  noting  the  resulting  group  of 
substitutions  of  the  elements  9?i,  ^'25  •  •  •  9m- 

If  /'i  is  a  self-conjugate  subgroup  of  T,  the  resulting  isomorphic 
group  H  will  be  regular,  as  is  easily  seen. 

§  101.     In  conclusion  we  deduce  the  following 

Theorem  XXXIII.  Given  any  number  of  mutually  mul- 
tiply isomorphic  groups,  in   ivhich  the   eleme7its  of   any  one  are 


GENEKAL    CLASSIFICATION    OF    GROUPS.  113 

all  different  from  those  of  any  other  one,  if  we  multiply  every  sub- 
titution  of  the  one  group  by  every  corresponding  substitution  of 
every  other  group  and  form  all  the  possible  products,  the  result 
is  an  intransitive  group,  and  conversely  every  intransitive  group 
can  be  constructed  in  this  way. 

The  first  part  of  the  theorem  is  sufficiently  obvious.  For  the 
second  part  we  consider  the  special  case  of  an  intransitive  group 
the  elements  of  which  break  up  into  two  transitive  systems.  The 
general  proof  is  obtained  in  a  perfectly  similar  way. 

Suppose  that  the  substitutions  sx  of  the  intransitive  group  G 
divide  into  two  components 

Sk—  (Tx'^K, 

where  ^a  affects  only  the  elements  Xi,X2,  ...  x,„ ,  and  r^  only 
^1?  ^25  •  •  •  ^lot-  It  is  possible  that  t^  occurs  also  in  other  combina- 
tions 

''"a'^A  ?     ^>J  A  5     ^k"^    A  J  .  •  • 

Similarly  r^  may  occur  in  other  combinations 

'^a'a?  ''■'a'^Aj  ''■"a^a  •  •  • 
We  coordinate  now  with  <>\  all  the  r^^  t\,  r\,  .  .  .  ,  and  with  t;^  all 
the  (Tx,  ^'a,  ^"a,  .  .  .  ,  and  proceed  in  the  same  way  with  all  the  sub- 
stitutions sx  of  G.  The  (7a's  form  a  group  -  and  the  r/s  a  group 
T.  Suppose  that  t;^,  ^r^  are  coordinated  with  "a,"^^-  Then  there 
are  substitutions  Sa  ,  s^ ,  s^ ,  such  that 

Sk  =  <yK^K<,   S^  =  (Tfj,r^, 

S\Sfj_  —  Si,  —  ^v'^v  ? 

and  consequently  ^^t^  =  a^  is  coordinated  with  r;^r^  =  r^ . 


CHAPTER  Y 


ALGEBRAIC  RELATIONS  BETWEEN  FTXCTIONS  BELONGING 
TO  THE  SAME  GROUP.  FAMILIES  OF  MULTIPLE- 
VALUED  FUNCTIONS. 

§  102.  It  has  been  shown  that  to  every  multiple-valued  func- 
tion there  belongs  a  group  composed  of  all  those  substitutions  and 
only  those  which  leave  the  value  of  the  given  function  unchanged. 
Conversely,  we  have  seen  that  to  every  group  there  correspond  an 
infinite  number  of  functions.  The  question  now  to  be  considered 
is  whether  the  property  of  belonging  to  the  same  group  is  a  funda- 
mentally important  relation  among  functions;  in  particular,  whether 
this  property  implies  corresponding  algebraic  relations. 

An  instance  in  point  is  that  of  the  discriminant  Jw,  of  the  values 
of  a  function  cr,  considered  in  Chapter  III,  §  55.  It  was  there  shewn 
simply  from  the  consideration  of  the  group  belonging  to  cr,  that  J^, 
and  therefore  the  corresponding  discriminant  of  any  function 
belonging  to  the  same  group,  is  divisible  by  a,  certain  power  of  the 
discriminant  of  the  elements  x^^x.,,  .  .  .  x„ . 

§  108.  We  shall  prove  now  another  mutual  relation  of  great 
importance. 

Theorem  I.  Two  functions  belonging  to  the  same  group 
can  he  rationally  expressed  one  in  terms  of  the  other. 

Suppose  <fi  and  v''i  to  be  two  functions  belonging  to  the  same 
group  of  order  r  and  degree  n 

G^i  =  [s,  =  1,  s,,S3,  .  .  .s,]. 

If  T.J  is  any  substitution  not  belonging  to  Gj ,  and  if  <f..  and  c\,  are  the 
values  which  proceed  from  cTi  and  c''i  by  the  application  of  <r.^ ,  then 
all  the  substitutions 

:also  convert  c,  and  c.^  iiito  v''i  and  ^''2  respectively,  and  these  are  the 
.onlv  substitutions  which  produce  this  effect.     The  values  V2  and  ^'2 


FUNCTIONS    BELONGING    TO    THE    SAME    GROUP.  115 

therefore  a^ain  belong  to  one  and  the  same  group  G2  =  (^2''^G-\'^i- 
Proceeding  in  the  same  way,  we  obtain  all  the  />  pairs  of  values  of 
(f  and  v's  together  with  the  p  corresponding  groups. 
For  every  integral  value  of  /,  the  function 

is  therefore,  like  9^1  +  f^-  -h  •  •  •  H-  9^p  or  v''i  +  4'-2  +  .  .  .  +  v'v?  ^^  iii^®- 
gral  symmetric  function  of  the  elements  Xy^x.,,  .  .  .  x„.  For  this 
function  is  merely  the  sum  of  all  the  values  which  (fx^4'\  can  assume 
and  is  accordingly  unchanged  by  any  substitution,  only  the  order 
of  the  several  terms  being  affected.  Accordingly,  if  (f^  and  0,  are 
integral  rational  functions  of  the  elements  x^,  then  A^  is  an  integral 
rational  function  of  Ci ,  c.2 ,  .  .  .  c„ . 

Taking  successively  /  =  0,  1,  2,  .  .  . />  —  1,  we   write    the  corres- 
ponding equations: 


.  .  ^''p,  every  v''a  is  obtained 


^I'^-'-rS  +  f^-V^  +  ^a^^V'sH- 
If  these  equations  are  solved  for  c'-, ,  6 
as  a  rational  function  of  ^1,  ^2?  •  •  •  S^p- 

§  104.  We  multiply  the  first  i>  — 2  equations  of  the  system  B) 
successively  by  the  undetermined  quantities  ^„,  2/1, 2/2  •  •  •  2/p-2)  and 
the  last  equation  by  ?/p_i  =  1,  and  add  the  resulting  products,  wri- 
ting for  brevity 

2/p-i9'-^""'-r2/p-29'"'"'  +  2/p-3^''~'+  .  .  .  H-^iC^  +  2/o=;^(^). 
We  obtain  then 

(1)     <\  '/,  (S^i)  -h  ^''2  /  (^2)  +  .  •  .  V^  Z  (s^p)  =  Ao2/o  +  AxVx  +  ^22/2  +  .  .  . 

.  .  •  ~r  -^p  -22/p  -  2  T~  -^p  -  i2/p  - 1  • 

From  this  equation  we  can  eliminate  ^''2?  ^''3,  •  •  •  v'v  and  obtain  ^''1. 
For  this  purpose  we  need  only  select  the  t/'s  so  that  we  have  simul- 
taneously 

■/Xfi)  =  ^,    ;?(fO  =  0,  ...;f(?p)  =  0;     z(n)  +  0. 


116  THEORY    OF     SUBSTITUTIONS. 

In  Chapter  III,  §  53,  we  have  shown  that  ^i,  ^2,  .  .  .  cTp  satisfy  an 
equation  of  degree  p 

the  coefficients  of  which  are  rational  in   c, ,  C2 ,  .  ,  .  c,, .     Again,  the 
quotient 

^(^)    :==(c._c.,)(^-c.3)...(c.-cg 


0  —  ip, 

vanishes  if  s^  =  ctj,  (p.>,  .  .  .  ^p.     But,  if  o  =  cTj,  we  have 

fe  —  ^2)^  —  ^3)  .  •  •  {^Pi  —  fp)  =  -^'fe)- 
The  derivative  X\(p^  is  not  zero,  for  if    a?,,  it'2,  .  .  .  ^„  are  independ- 
ent, the  values  ^1,  ^^2)  •  •  •  S^p  are  all  different. 

We  can  therefore  satisfy  the  requirements  above  by  taking 

that  is, 

yp-2  =  — A,         yp-S=i%,         2/p-4  =  —  ^4,    •   .   .    2/0=    ±i^"p- 

Or,  if  we  write 
we  have 

and  consequently 

^p  -  2  ==  ^1  —  «n  Up  -  3  =  fi'—  «i  '7^1  +  '^2 ,  .?/p  -  4  =  V^'—  ^^-1  t^i'"+  'h'Pi  —  «3 ,  •  • . 
By  substitution  in  (1)  we  obtain  then 

(2)  ,',Xfe)=«fe),     ^'■■  =  I^J- 

The  value  of  ^''i  thus  obtained  can  be  reduced  to  a  simpler  form 
as  follows.     The  product 

X'{v,)X'{<p,)...X'{^,) 

is  a  symmetric  function  of  the  c?'s,  and  in  fact,  as  appears  from  the 
expression  for  X\(pi)  above,  only  differs  from  the  discriminant  J^ 
in  algebraic  sign.     Moreover,  the  product 

(3)  r{<f,)r{<p,)...x\<fp) 

is  a  symmetric  function  of  the  roots  of  the  equation 


FUNCTIONS    BELONGING    TO    THE    SAME    GBOUP.  117 


X{<p) 


0, 


9  —  <Pi 
and  can  therefore  be  rationally  expressed  in  terms  of  the  coefficients 
of  this  equation,  that  is,  in  terms  of  «!,  "25  •  •  •  "p?  and  <p^,  and  con- 
sequently in  terms  of  Ci,c.2,  .  .  .  c^  and  <Pi .  If  now  we  multiply 
numerator  and  denominator  of  the  expression  for  (/'i  in  (2)  by  the 
product  (3),  we  obtain 

(4)  .'.  =  ^. 

The  denominator  of  this  last  fraction  is  rational  and  integral  in 
Cj,  C2,  .  .  .  c„;  the  numerator  is  rational  and  integral  in  Cj,  C2,  .  .  .  c„ 
and  9?i. 

If  the  numerator  -Ri(^i)  is  of  a  degree  higher  than  />  —  1  with 
respect  to  <Pi,  a  still  further  reduction  is  possible.  For  suppose 
that 

R^{<p)  =  X{<p)Q{<p)  +  R,{<p), 

where  Q{<p)  and  B^i^)  are  the  quotient  and  remainder  obtained  by 
dividing  Ri{(p)  by  X{(p).     The  degree  of  Rii^p)  then  does  not  exceed 

P — 1.      Now  if  <p  =  <Pi,<P2,  '  '  ■  9p,  ^W)  —  ^'     Consequently 

R,{<Pk)=^R-2{<Pk)     (A  =  1,2,3,.../>), 
and  therefore 

Similar  considerations  hold  for  the  values  4'2i4'i,  -  -  -  4'^-  We 
have  therefore 

Theorem  II.  If  two  p -valued  functions  (fx  and  4'k  belong  to 
the  same  group  G\ ,  theii  (/'\  can  be  expressed  as  a  rational  f miction 
of  which  the  denominator  is  the  discriminant  J^  and  is  therefore 
rational  and  integral  in  Cj,  Cg,  .  .  .  c„,  ivhile  the  numerator  is  an 
integral  rational  function  of  ^x,  of  a  degree  not  exceeding  p  —  1, 
with  coefficients  which  are  integral  and  rational  in  Cj,  C2,  .  .  .  c„* 

§  105.     The  converse  of  Theorem  I  is  proved  at  once : 

Theorem  III.  If  two  functions  can  be  rationally  ex- 
pressed one  in  terms  of  the  other ^  they  belong  to  the  same  group. 

*Cf.  Kronecker:  Cre]le9l,  p.  307. 


118  THEORY    OF    SUBSTITUTIONS. 

In  fact,  given  the  two  equations 

it  appears  from  the  former  that  (p  is  unchanged  by  all  substitutions 
which  leave  i!<  unchanged,  so  that  the  group  of  cp  contains  that  of  9'', 
while  from  the  latter  equation  it  appears  in  the  same  way  that  the 
group  of  v''  contains  that  of  <p.  The  two  groups  are  therefore  iden- 
tical. 

Remark.  Apparently  the  proof  of  this  theorem  does  not  involve 
the  requirement  that  (p  and  4'  shall  be  rational  functions.  It  must 
however  be  distinctly  understood  that  this  requirement  must  always 
be  fulfilled.     For  example,  in  the  irrational  functions 


-2  > 


^^  X-y  OCyOC^  ~\    3C2  J     '\'  OCy  LiOCy)C2    \~  X"2  ,     '\'  X-y   ~j~  ciX-ytlC2  ~t~  3C2 

the  expressions  under  the  square  root  sign  are  all  unchanged  by 
the  transposition  (7  —  {x-^x^.  But  it  remains  entirely  uncertain 
whether  the  algebraic  signs  of  the  irrationalities  are  affected  by  this 
substitution.  Considerations  from  the  theory  of  substitutions  alone 
cannot  determine  this  question,  and  accordingly  the  sphere  of  appli- 
cation of  this  theory  is  restricted  to  the  case  of  rational  functions. 
If,  in  the  last  two  irrationalities  above,  the  roots  are  actually 
extracted  and  written  in  rational  form 

31  yXi        XiJ^        IE  \X\    |~  «^2/? 

it  appears  at  once  that  the  transposition  <7  changes  the  sign  of 
the  former  expression  but  leaves  that  of  the  latter  unchanged, 
while  in  the  case  of  the  first  irrationality  this  matter  is  entirely 
undecided. 

§  106.  Theorems  I  and  III  furnish  the  basis  for  an  algebraic 
classification  of  functions  resting  on  the  theory  of  groups.  All 
rational  integral  functions  which  can  be  rationally  expressed  one 
in  terms  of  another,  that  is,  which  belong  to  the  same  group,  are 
regarded  as  forming  a  family  of  algebraic  functions.  The  number 
f)  of  the  values  of  the  individual  functions  of  a  .family  is  called  the 
order  of  the  family.  The  several  families  to  which  the  different 
values  of  any  one  of  the  functions  belong  are  called  conjugate 
families.''^ 

♦L.  Kronecker:  Monatsber.  d.  Berl.  Akad.,  1879,  p.  212. 


FUNCTIONS    BELONGING    TO    THE    SAME    GROUP.  119 

The  product  of  the  order  of  a  family  by  the  order  of  the  cor- 
responding group  is  equal  to  7il,  ivhere  n  is  the  degree  of  the  group. 

Every  function  of  a  family  of  order  ft  is  a  root  of  an  equation 
of  degree  p,  the  coefficients  of  which  are  rational  in  Cj,  C2,  .  .  .  c,^. 
The  remaining  />  —  1  roots  of  this  equation  are  the  conjugate  func- 
tions. 

The  groups  which  belong  to  conjugate  fdmilies  have,  if  i>  >  2, 
n>  4:,  no  common  substitution  except  the  identical  substitution. 

For  p  =  2  the  tivo  conjugate  families  are  identical. 

For  p  =  Q,  n  —  4t  there  is  a  family  which  is  identical  with  its 
five  conjugate  families. 

§  107.  In  the  demonstration  of  §  104  the  condition  that  <p  and 
(/'  should  belong  to  the  same  family  was  not  wholly  necessary.  It  is 
only  essential  that  </'  shall  remain  unchanged  for  all  those  substitu- 
tions which  leave  the  value  of  <p  unaltered.  The  demonstration 
would  therefore  still  be  valid  if  some  of  the  values  of  0  should 
coincide;  but  the  values  of  (p  must  all  be  different,  as  appears,  for 
example,  from  the  presence  of  the  discriminant  J^  in  the  denomi- 
nator of  (r'.  Under  the  more  general  condition  that  the  group  of 
(p  includes  that  of  ■  s^  we  have  then  the  following 

Tlieoreni  IV.  //  a  function  (p  is  unchanged  by  all  the  sub- 
stitutions of  the  group  of  a  second  function  <p,  ivhile  the  converse  is 
not  necessarily  true,  then  0  can  be  expressed  as  a  rational  function 
of  (p,  as  in  Theorem  II. 

Under  these  circumstances  the  family  of  the  function  (p  is  said 
to  be  included  in  the  family  of  the  function  (p.  4'  can  be  rationally 
expressed  in  terms  of  ^,  but  <p  cannot  in  general  be  thus  expressed 
in  terms  of  4.>.  An  including  group  corresponds  to  an  included  fam- 
ily and  vice  versa.  The  larger  the  group  the  smaller  the  family, 
the  same  inverse  relation  holding  here  as  between  the  orders  r 
and  p. 

From  the  preceding  considerations  we  further  deduce  the  fol- 
lowing theorems: 


120  THEORY    OF    SUBSTITUTIONS. 

Theorem  V.  It  is  always  possible  to  find  a  function  in 
terms  of  which  any  number  of  given  functions  can  be  rationally 
expressed.  This  function  can  be  constructed  as  a  linear  combina- 
tion of  the  given  functions.  Its  family  includes  all  the  families 
of  the  given  functions. 

Thus  any  given  functions  9?,  0,  Xi  •  •  •  can  be  rationally  expressed 
in  terms  of 

where  «,  /5,  ^,  .  .  .  are  arbitrary  parameters.  For  the  group  of  m  is 
composed  of  those  substitutions  which  leave  <p,  4'iy,,  •  -  -  all  un- 
changed, and  which  are  therefore  common  to  the  groups  of 
^ ,  0 ,  / ,  .  .  .  The  group  of  w  is  therefore  contained  in  that  of 
every  function  ^ ,  0 ,/,...  ,  and  the  theorem  follows  at  once. 

A  special  case  occurs  when  the  group  of  w  reduces  to  the 
identical  operation,  co  being  accordingly  a  n!-valued  function.  In 
this  case  every  function  of  the  n  elements  x^^x.^^  .  .  .  x,,  can  be 
rationally  expressed  in  terms  of  w,  and  every  family  is  contained 
in  that  of  lo.     The  family  of  o)  is  then  called  the  Galois  family. 

Theorem  VI.  Every  rational  function  of  n  independent 
elements  x^^x^,  .  .  .x^  can  be  rationally  expressed  in  terms  of  every 
nUvalued  function  of  the  same  elements;  in  particular,  in  terms 
of  any  linear  function 

where  «i ,  «2  ?  •  •  •  "«  ^^'^  arbitrary  parameters. 

§  108.  We  attempt  now  to  find  a  means  of  expressing  a  mul- 
tiple-valued function  <p  in  terms  of  a  less  valued  function  <',  the 
group  of  the  former  being  included  in  that  of  the  latter.  A  rational 
solution  of  this  problem  is,  from  the  preceding  developments,  im- 
possible. The  problem  is  an  analogue  and  a  generalization  of  that 
treated  in  Chapter  III,  §  53,  where  a  />-valued  function  was'^l 
expressed  in  terms  of  a  symmetric  function  by  the  aid  of  an  equa- 
tion with  symmetric  coefficients  of  which  the  former  was  a  root.  \ 

From  the  analogy  of  the  two  cases  we  can  state  at  once  the 
present  result: 


FUNCTIONS    BELONGING    TO    THE    SAME    GROUP.  121 

Theorem  VII.     If  the  group  of  a  mp-valued  function  <p  is 
contained  in  that  of  a  p-valued  function  </',  and  if 

are  the  m  values  which  (p  takes  under  the  application  of  all  the 
substitutions  which  leave  4'  unchanged,  then  these  m  values  of  <p  are 
the  roots  of  an  equation  of  degree  m,  the  coefficients  of  which  are 
rational  functions  of  ^''. 

In  fact,  the  substitutions  of  the  group 

Gi  =  [si  =  l,s,,,S3,  .  .  .s,.]      'i'^-j 

of  01  are  applied  to  any  symmetric  function  of  ^i,  ^2?  •  •  •  9m,  the 
value  of  this  function  is  unchanged,  only  the  order  of  the  several 
terms  being  altered.  In  particular  we  have  for  the  elementary 
functions 

<Pi-i-<P'2     -h  .  .  .  +  <Pm  —  ^1  (0i), 

^1^2  +  9x9^  +   .   .   .    +  9m-x9.n  —  ^M^, 


<Px<Pl  .   .   .  9m  =  A^{(/\\ 

where  the  A's  are  rational,  but  in  general  not  integral  functions  of 
V^'i .     We  obtain  therefore  the  equation 

(A,)  ^-_-A,(vS)^"'-^  +  A2(0O^'"-'+  .  .  •  ±  A.{</'x)=0, 

of  which  the  roots  are  ^i ,  9^2  >  •  ♦  •  9m  5  and  in  general  the  equation 

(A,)  9'"— A,  (</'^)<p'"-'-{-  A,{</'^)<p—'+  .  .  .  ±  A,(v^'x), 

of  which  the  roots  are  the  m  values  of  ^  which  correspond  to  the 

value  0A  of  (/'. 

The  denominators  of  the  A^'s  and,  in  fact,  their  least  common 
denominator  is  always  a  divisor  of  the  discriminant  J^,  as  appears 
from  the  proof  of  Theorem  II.  If  (/>  is  a  symmetric  function,  there 
is  no  longer  a  discriminant,  and  the  denominator  is  removed,  as  we 
have  seen  in  Chapter  III,  §  53. 

§  109.  One  special  case  deserves  particular  notice.  If  the 
included  group  H  of  the  function  <p  is  commutative  with  the  inclu- 
ding group  G  of  (/'  then,  if  a  single  root  of  the  equation  (Aj)  is 
known,  the  other  roots  are  all  rationally  determined.     For  if 

9X,   92,  9i,   •  -   •  9rH 


that 

is, 

• 

G.-'H,n,= 

■H„ 

''■3     H^(T^  — 

H, 

and 

consequently 

H, 

—  H2  =  Hs 

=    . 

122  THEORY    OF    SUBSTITUTIONS. 

are  the  roots  of  the  equation  (Aj),  and  if 

ifj,  H.,,  i?3,  .  .  .  H,„ 
are  the  groups  belonging  to  these  values,  finally,  if 

Tj  =  1,  'T-.,,  <T3  .  .  .  fT,„ 

are  any  arbitrary  substitutions  of  G  which  convert  <Pi  into  ^1 ,  ^2  ?  •  •  •  ^m 
respectively,  then  we  have  (Chapter  III,  §  45) 

H,  =  rT,-'H,rT„  H,=  <T,-'H,<T,,...H„,  =  ^-'H,<7,„. 

But  by  supposition  jH"  is  a  self -conjugate  subgroup  of  G,  and  there- 
fore 

G~  Hi  G  =  Hi , 

.  .    fT„-^HiT„^  —  Hi, 

The  m  different  values  Vi,  s^2>  •  •  •  ^»,  therefore  belong  to  one  and  the 
same  group  if,  and  can  consequently  all  be  rationally  expressed  in 
terms  of  any  one  among  them,  in  accordance  with  Theorem  I. 

The  family  of  c''i  is  included  in  that  of  <pi .  When,  as  in  the 
present  case,  the  group  H^  of  v'l  is  not  merely  contained  in  the 
group  G  of  v''i  but  is  a  self- conjugate  subgroup  of  G,  the  family  of 
(/\  is  called  a  self- conjugate  subfamily  of  the  family  of  c^i . 

Theorem  VIII.  In  order  that  all  the  roots  of  the  equation 
(Aj)  should  be  rationally  expressible  in  terms  of  any  one  among 
them,  as  <pi ,  it  is  necessary  and  suj^flcient  that  the  family  of  v''i 
should  be  a  self-conjugate  subfamily  of  that  of  v'j,  i.  e.,  that  the 
group  of  <Pi  should  be  a  self- conjugate  subgroup  of  that  of  ^i-  The 
groups  of  (pi,  <p.2i  •  •  •  <Pm  «^'«  then  coincident. 

We  consider  in  particular  the  case  where  m  is  a  prime  number. 
Suppose  G-i  to  be  the  group  of  C\  and  H^  that  of  (fi .  Since  every 
substitution  of  G^  produces  a  corresponding  substitution  of  the  val- 
ues v'l  J  ^2  J  •  •  .  Vw  5  the  group  G^  is  isomorphic  with  a  group  of  the 
S^'s.  The  latter  group  is  transitive  and  of  degree  m.  From  The- 
orem II,  Chapter  IV,  its  order  is  divisible  by  m,  and  from  Theorem 
X,  Chapter  III,  it  therefore  contains  a  substitution   of  order   m. 


FUNCTIONS  BELONGING  TO  THE  SAME  GROUP. 


12a 


For  m  elements,  where   m  is  prime,  there  is  only  one  type  of  such 
substitutions 

The  corresponding  substitution  r  of  G^  therefore  permutes 
?'i )  V2  5  •  •  •  ^>n  cyclically.  Moreover,  since  r"*  corresponds  to  t"\  it  fol- 
lows that  r'"'  leaves  all  the  functions  9^1 ,  ^^2  >  •  •  •  9>n  unchanged.  Ac- 
cordingly r'",  and  no  lower  power  of  r,  is  contained  among  the  sub- 
stitutions of  H^ . 

Furthermore  we  readily  show  that 

For  the  substitutions  ifi,  iJjr,  .  .  .  H^t"'~^  are  all  different  and,  since 

H,  is  of  order  — '- ,  there  are  m  — -  =  — ^  of  them.     They  are  all  nec- 
mp  mp       f) 

Til 

essarily  contained  m   Gi,  which,    being  itself  of  order  — ,  cannot 

contain  any  other  substitutions.     From  this  it  appears  again  that  r 
is  commutative  with  H^ . 

Theorem  IX.  If  the  equation  {A^  is  of  prime  degree  my 
and  if  the  group  H^  of  (p^  is  a  self-conjugate  subgroup  of  the 
group  Gi  of  v''ij  then  G^  contains  a  substitution  r  ivhich  permutes 
fi  5  S^2?  '  ■  '  <Pp  cyclically.  This  substitution  is  commutative  with 
Hi ;  its  m^^,  and  no  loiver,  power  is  contained  in  H^ ;  together  with 
Hy  it  generates  the  group  G\ . 

§  110.  We  examine  now  under  what  circumstances  (Aj)  can 
become  a  binomial  equation,  again  assuming  the  degree  m  to  be  a 
prime  number.  If  {A^  is  binomial,  its  roots,  (f-^,  w^j,  w'Vi,  .  .  .f'^"'"  Vi 
evidently  all  belong  to  the  same  group.  It  is  therefore  necessary 
that  6ri  should  be  a  self- conjugate  subgroup  of  G^. 

We  proceed  now  to  show  conversely  that,  if  the  group  H^  is  a 
self- conjugate  subgroup  of  G, ,  then  a  function  /i  belonging  to  H^ 
can  be  found,  the  m*^  power  of  which  belongs  to  Gi . 

Denoting  any  primitive  m^^  root  of  unity  by  <y,  we  write 

7a  =  Vl  +  «>9^2  +  ^Vs  +   •  .  •  +«>'""  Vm- 

If  we  apply  to  this  expression  the  successive  powers  of  t  or  r,  we 
obtain 


124 


THEORY    OF    SUBSTITUTIONS. 


/2  =  9''2  +  "^S^3  +  ^V4+   .  .  .    +^"'      Vl  =  <^~V 


y.2  —  <T%  +  w^4  +  ^Vr.  +  . . .  4-  «>"•" V2  =  ^'^~ 7 


15 


H 


■consequently 

/I      —  /2      —    .  .        —  /«,      . 

We  have  now  to  prove  1)  that  /^  belongs  to  the  group  if  1 ,  and 
2)  that  7i"'  belongs  to  the  group  G^ . 

In  the  first  place,  since  ^i ,  9^2  ?  •  •  •  ^^  are  unchanged  by  all  the 
substitutions  of  H^ ,  the  same  is  true  of  /i .  Moreover  if  there  were 
any  other  substitutions  which  left  /,  unchanged  we  should  have,  for 
example, 

and  therefore 

The  latter  equation  would  then  have  one  of  its  roots,  and  conse- 
quently all  its  roots,  in  common  with  the  irreducible  equation 

and  we  must  therefore  have 

But  we  may  assume  the  function  (f-^  to  have  been  constructed  by  the 

n\ 
method  of  §  31  as  a  sum  of  — '-  terms  of  the  form  x.'^x.j^  .  .  .    with 
^  mp 

undetermined  exponents.  The  systems  of  exponents  in  9^1 ,  f  2  ?  •  •  •  9>n 
will  then  all  be  different,  and  therefore,  since  the  a^'s  are  independ- 
ent variables,  the  equation 

•can  hold  only  if  (f^  ^  (p/^  and  9?^  =  ^,.,  identically.  The  function  /, 
therefore  belongs  to  H^. 

It  follows  at  once  that  /i'"  belongs  to  G^  •  For  this  function  is 
unchanged  by  H^  and  r,  and  consequently  by 

G,=  \H,,r\. 

No  other  substitutions  can  leave  x^"  unchanged.  For  otherwise 
X"*  would  take  less  than  />  values,  and  its  m^^  root  Xi  l^ss  than  inp 
Talues,  which  would  be  contrary  to  the  result  just  obtained. 


FUNCTIONS    BELONGING    TO    THE    SAME    GROUP.  125 

Tlieoreill  X.  In  order  that  the  family  belonging  to  a  group 
H  may  contain  functions  the  m^^^  power  of  which  belongs  to  the 
family  of  a  group  G,  it  is  necessary  and  sufficient  that  H  should  be 
a  self- conjugate  subgroup  of  G,  or,  in  other  words,  that  the  famil'y 
of  G  should  be  a  self- conjugate  subfamily  of  that  of  H. 

From  Theorems  IX  and  X  the  following  special  case  of  the  lat- 
ter is  readily  deduced : 

Theorem  XI.  In  order  that  the  prime  power  (f^'  of  a  pp- 
valued  function  cp  may  have  />  values,  it  is  necessary  and  sufficient 
that  there  should  be  a  substitidion  r,  commutative  with  the  group  H 
of  ip,  of  which  the  p^^  power  is  the  first  to  occur  in  H. 

Finally  an  extension  of  the  last  theorem  furnishes  the  following 
important  result: 

Theorem  XII.     If  the  series  of  groups 
G,  Gi,  G2,  G3,  .  .  .  Gv 

is  so  connected  that  every  Ga-i  can  be  obtained  from  the  following 
Ga  by  the  addition  to  the  latter  of  a  substitution  r^  commutative 
with  Ga,  of  ivhich  a  prime  i^ower,  the  pj^^,  is  the  first  to  occur  in  Gay 
then  and  only  then  it  is  possible  to  obtain  a  i>  - Pi  ■  Pi .  .  .  Pv-valued 
function  belonging  to  G^  from  a  [> -valued  function  belonging  to  G  by 
the  solution  of  a  series  of  binomial  equations.  The  latter  are  then 
of  degree    Pi,P2,2h,  •  •  -Pvj  respectively. 

§  111.  In  the  expression  of  a  given  function  in  terms  of 
another  belonging  to  the  same  family,  we  have  met  with  rational 
fractional  forms  the  denominators  of  which  were  factors  of  the  dis- 
criminant of  the  given  function.  If  we  regard  the  elements 
x^,  X.2,  .  .  .  X,,  as  independent  quantities,  as  we  have  thus  far  done, 
the  discriminant  of  any  function  ^  is  different  from  zero,  for  the 
various  conjugate  values  of  c^  have  different  forms.  But  if  any 
relations  exist  among  ihe  elements  x,  it  is  no  longer  true  that  a  dif- 
ference in  form  necessarily  involves  a  difference  in  value.  It  is 
therefore  quite  possible  that  if  the  coefficients  in  the  equations 

f[x)  =  X" C-iX"~'^  -\-  CoX""'^ —  ...  ±  c„  =  0 

are  assigned  special  values,  the  discriminant 


126  THEORY    OF    SUBSTITUTIONS. 

may  become  zero.  If  this  were  the  case,  c>  could  not  be  employed 
in  the  expression  of  other  functions  of  the  same  family.  And  it  is 
conceivable  that  the  discriminant  of  every  function  of  a  family 
might  vanish.  It  is  therefore  necessary,  in  order  to  remove  this 
uncertainty,  to  prove 

Theorem  XIII.  If  only  no  Uvo  x^s  are  equal,  then  ivhatever 
other  relations  may  exist  among  the  x^s,  there  are  in  every  family 
functions  the  discriminants  of  which  do  not  vanish. 

A  proof  might  be  given  similar  to  that  of  §  30.  It  is  however 
more  convenient  to  make  use  of  the  result  there  obtained,  that 
under  the  given  conditions  there  are  still  ti!- valued  functions  of  the 

form 

if  =  «(,  -\-  a^x^  +  a,^x.2^  +  .  .  .  +  «„^„ . 

We  suppose  the  a's  and  the  .r's  to  be  free  to  assume  imagin- 
ary (complex)  as  well  as  real  values.  This  being  the  case,  if  the 
n\  values  of  <f  are  all  different,  we  can  select  the  coefficient  «„,  so 
that  the  moduli  of  the  values  of  <p  are  also  all  different.     For  if 

<fk  =  mx  +  Px  V  —  1       (/  =  1,  2,  .  .  .  w), 
then  we  can  take 

in  such  a  way  that  all  the  n !  quantities 

^''A  =  Va  +  «'=  (mxi-p)  +  (/^A  -{-q)^ZZi     (/  =  1,  2,  .  .  .  n) 
shall  have  different  moduli.     For  from 

{m,  +p)-^+  (/.,  +  qY  =  K  +  P)'+  {!'k  +  qf 
it  would  follow,  if  p  and  q  are  entirely  arbitrary,  that 

mx  =  m^,  !J.K  =  !J-K- 
In  fact  we  can,  for  example,  take  p  =  q'  and  q  so  large  that  even 
special  values  of  q  satisfy  the  conditions. 

Suppose  then  that  the  4'^^  are  arranged  in  order  of  the  magni- 
tudes of  their  moduli 

<'x ,  <'i  J  <'^ ,  •  .  •  <'u\     (mod.  4'K  >  mod.  v''a  + 1)  • 
We  take  then  the  integer  e  so  great  that 

vv >  (v''A+r+^''A+/+ . . .  +  v''.r)   {^^  =  1, 2, . . .  (^i!-i)). 


THE    NUMBER    OF    VALUES    OF    INTEGRAL    FUNCTIONS.  127 

From  every  equation  of  the  form 

n  4': + vs;  +  <':  + . .  •  =  4': + 0/  +  ^\^  + . . . 

it  follows  accordingly  that  a  =  « ,  6  =  /5 ,  c  =  ^ ,  .  .  .  If  now  we  apply 
the  r  substitutions  of  G  to  4'\%  and  add  the  results,  the  sum 

is  a  function  of  the  required  kind.  For  in  the  first  place  w  is  evi- 
dently unchanged  by  G.  And  in  the  second  place  the  properties  of 
the  equation  't')  show  that  o>  has  />  distinct  values,  and  consequently 
Ja>  is  not  zero. 


CHAPTEE  VI. 


THE  NUMBER  OF  VALUES  OF  INTEGRAL  FUNCTIONS. 

§  112.  Thus  far  we  have  obtained  only  ocasional  theorems  in 
regard  to  the  existence  of  classes  of  multiple-valued  functions.  We 
are  familiar  vpith  the  one-  and  two-valued  functions  on  the  one  side 
and  the  7i! -valued  functions  on  the  other.  But  the  possible  classes 
lying  between  these  limits  have  not  as  yet  been  systematically  exam- 
ined. An  important  negative  result  was  obtained  in  Chapter  III, 
§  42,  where  it  was  shown  that  />  cannot  take  any  value  which  is  not 
a  divisor  of  n\.  Otherwise  no  general  theorems  are  as  yet  known 
to  us.  We  can,  however,  easily  obtain  a  great  number  of  special 
results  by  the  construction  of  intransitive  and  non-primitive  groups. 
But  these  are  all  positive,  while  it  is  the  negative  results,  those 
which  assert  the  non-existence  of  classes  of  functions,  that  are  pre- 
cisely of  the  greatest  interest. 

The  general  theory  of  the  construction  of  intransitive  groups 
would  require  as  we  have  seen  in  §  101,  a  systematic  study  of  iso- 
morphism in  its  broadest  sense.  We  shall  content  ourselves  there- 
fore with  noting  some  of  the  simplest  constructions. 

Thus,  if  there  are  n  =  a-\-b-\-G-\-  .  .  .  elements  present,  and 
if  we  form  the  symmetric  or  the  alternatiog  group  of  a  of  them, 
the  symmetric  or  alternating  group  of  b  others,  and  so  on,  then  on 
multiplying  all  these  groups  together,  we  obtain  an  intransitive 
group  of  degree  n  and  of  order 

r  =  £  a!  bid  .  .  ., 

where  £=:l,J,i,J,...,  according  as  the  number  of  alternating 
groups  employed  in  the  construction  is  0,  1,  2,  3,  ...  ,  the  rest  being 
all  symmetric.  For  the  number  of  values  of  the  corresponding 
functions  we  have  then    . 

?i! 

P  = 


salblel  .  .  . 


THE    SfUMBEE    OF    VALUES    OF   INTEGRAL    FUNCTIONS.  129 

By  distributing  n  in  different  ways  between  a,  6,  c,  . . . ,  we  can 
obtain  a  large  number  of  classes  of  functions.  For  example,  if 
n  —  5,  we  may  take 

a  =  5;  £  =  J,  /^=    2;   <P2  =  {xx—x^){x^—x^)...{x^—x^) 

a  ==  4,  &  =  1 ;  £  =r  1,  ^  =    5 ;   ^3  =  x^x^x^x^. 

a  =  4,  6=1;  £  =  i    ^  =  10;    ^4  =  (a?i  —  x^{x^  —  x^{xy^  —  x^ 

[X^       a^a)  (^2       X^)  {X-i       X^ . 
a  =  3,  6  =  2 ;  £  =  1,  />  =  10;   ^5  =  x^x^x^ -\- x^Xr^, 

a  =  3,  6  =  2;  e  =  1    ^  =  20;    ^e  =  {x^  —  x^) (x^  —  x-^) {x^  —  x^) 

\    X^Xt^. 

ip^  —  x^x<fc^-\rx^—Xr,, 
a  =  3,  &  =  2;  £  =  4,  ^>  =  40;   s^8  =  {x^  —  x^) {x^  —  x^ {x^  —x^ 

\  x^      a?5 . 
a  =  3,  6  =  1,  c  =  1 ;  £  =  1,  />  =  20;    ^^  =  a^ja^gO^a. 

The  imprimitive  groups  give  rise  in  a  similar  way  to  the  con- 
struction of  functions  with  certain  values.  For  example,  for  n  —  ^^ 
we  may  take  any  two  systems  of  non-primitivity  of  three  elements 
each,  or  any  three  systems  of  two  elements  each,  and  with  these 
construct  various  groups,  the  theory  of  which  depends  only  on  that 
of  groups  of  degrees  two  and  three. 

§  113.  General  and  fundamental  results  are  not  however  to  be 
obtained  in  this  way.  We  approach  the  problem  therefore  from  a 
different  side,  which  permits  us  to  give  it  a  new  form  of  statement. 

Oiven  a  />-valued  function  ^1  with  a  group  G^ ,  we  construct  again 
the  familiar  table  of  §  41 : 


^1; 

Sj  —  1,     §2,          S3,          .   , 

.Sr    ; 

G^ 

n\ 

'^2)                S2<r2,     SgC-^,    .   . 

,S,n^', 

G,<T, 

^3; 

'^3,               ^2^3,     Sa^Tg,    .  . 

.S,<ra; 

G^(T^ 

We  proceed  then  to  examine  the  distribution  of  the  substitutions  of 
a  given  type  among  the  lines  of  this  table. 

A)     There  are  n  —  1  transpositions    {x^x„),  («  =  2,  3, , .  .  n).    If 


130 


THEORY    OF    SUBSTITUTIONS. 


then  ()  <  n^  and  if  the  group  G^  of  (f-^  does  not  contain  any  trans- 
position of  the  form  {x-^x^,  these  {n — 1)  transpositions  are  distribu- 
ted among,  at  the  most,  {n — 2)  lines  of  the  table.  Accordingly  some 
line  after  the  first  must  contain  at  least  two  of  them.  Suppose  these 
two  are 

Then  it  appears  that  a  combination  of  the  two 

{X^X„X^)  —  {x^Xa)  {XiX^)  =  Sa<Tx{Sp<rK)-^  =  Sa^K^r'Sp"'  =  SaSp~^  =  Sy 

occurs  in  Gi .  Consequently,  if  />  <  n,  Gi  contains  either  a  transpo- 
sition or  a  circular  substitution  of  the  third  order,  including  in 
either  case  a  prescribed  element  x^ .  The  same  is  obviously  true  of 
Any  prescribed  element  X\. 

B)  There  are     ^    ^ — ^  transpositions  of  the  form  (Xaa?^),  (a  == /J 

=  1,  2,  .  . .  n).     If  therefore  p  <_        n —  >  ^^^^  if  the  first  line  of  the 

table  does  not  contain  any  transposition,  then  some  other  line  con- 
tains at  least  two.  If  these  have  one  element  in  common,  as  {x^x^^ 
{XgX^^  then,  as  we  have  seen  in  A),  their  product  {XgX^Xy)  occurs  in 
(ti  .  If  they  have  no  element  in  common,  as  {x^x^,  (XyXs),  then 
their  product  (XaXp)  (XyXs)  also  occurs  in  Gi .  In  either  case  Gi 
therefore  contains  a  substitution  of  not  more  than  four  elements. 

C)  There  are  (n  —  1)  (n  —  2)  substitutions  of  the  form  (x^XaXp), 
(a  J=/?  =  2,  3,  .  .  .  n).  If  therefore  r  <{n  —  1)  (n  —  2),  and  if  6?i  con- 
tains no  substitution  of  this  form,  some  other  line  of  the  table  con- 
tains at  least  two  of  them.  A  combination  of  these  shows  that  Gi 
contains  substitutions  which  affect  three,  four,  or  five  elements. 

Proceeding  in  this  way,  we  obtain  a  series  of  results,  certain 
of  which  we  present  here  in  the  following 

Theorem  I.  1)  //  the,  number  p  of  the  values  of  a  function 
is  not  greater  than  n — 1,  the  group  of  the  function  contains  a  sub- 
stitution off  at  the  most,  three  elements,  including  any  prescribed 

element.     2)     //  />  is  not  greater  than        ^ — -,  the  group  of  the 

function  contains  a  substitution  of,  at  the  most,  four  elements.     3) 

If  pis  not  greater  than— ^ -,  the  group  of  the  function 


THE    NUMBER    OF    VALUES    OF    INTEGRAL    FUNCTIONS.  131 

contains  a  substitution  of,  at  the  most,  six  elements.     ^  If  p  is  not 

.     .u      n{n  —  l){n—2)...(n—k-{-l)    .,        ^  ...      . 

greater  than  — — j^ ^ — -,  the  group  of  the  func- 

tion  co7itains  a  substitution  of,  at  the  most,  2k  elements,  b)  If  p  is 
not  greater  than  {n — 1)  (n — 2)  .  .  .  {n — k-\-l),  the  group  of  the 
function  contains  a  substitution  of,  at  the  most,  2k — 1  elements, 
including  any  prescribed  element,  so  that  the  group  contains  at  least 

^r- such  substitutions. 

2k — 1 

By  the  aid  of  these  results  the  question  of  the  number  of  values 
of  functions  is  reduced  to  that  of  the  existence  of  groups  contain- 
ing substitutions  with  a  certain  minimum  number  of  elements. 

§  114  In  combination  with  earlier  theorems,  the  first  of  the 
results  above  leads  to  an  important  conclusion. 

From  Chapter  IV,  Theorem  I,  we  know  that  the  order  of  an 
intransitive  group  is  at  the  most  {n  —  1)!.  Consequently,  the  num- 
ber of  values  of  a  function  with  an  intransitive  group  is  at  least 

n ' 

-. -rr-  =  n.  For  such  a  function  therefore  />  cannot  be  less  than  n. 

(n— 1)! 

Again,  the  order  of  a  non-primitive  group  is,  at  the  most,  2 !  I  -^ !  I  , 

so  that  the  number  of  values  of  a  function  with  a  non-primitive 

n\ 
group  is  at  least '- .     For  n  =  4,  this  number  is  less  than  n; 

2'— 1— » 
^•2     2' 

but  for  w  >  4,  it  is  greater  than  n.  For  such  a  function  then,  if 
n  >  4,  ^  cannot  be  less  than  n.  Again  for  the  primitive  groups  it 
follows  from  Chapter  IV,  Theorem  XVIII,  in  combination  with  the 
first  result  of  Theorem  I,  §  113,  that  if  p<n,  the  corresponding 
group  is  either  alternating  or  symmetric,  that  is,  /?  =  2  or  1.  The 
non-primitive  group  for  which  ii  =  4,  ^  =  4,  r  =  8  is  already  known 
to  us,  (§  46).     We  have  then 

Theorem  II.  If  the  number  p  of  the  values  of  a  function  is 
less  than  n,  then  either  p  =  l  or  p  =  2,  and  the  group  of  the  func- 
tion is  either  symmetric  or  alternating.  An  exception  occurs  only 
for  n  =  4:,  p  =  S,  r  =  8,  the  corresponding  group  being  that  belong- 
ing to  XiX2-\-x^x^. 


132  THEORY    OF    SUBSTITUTIONS. 

§  115.     On  account  of  the  importance  of  the  last  theorem  we 
add  another  proof  based  on  different  grounds.  * 
Suppose  ^  to  be  a  function  with  the  p  <n  values 

If  we  apply  any  substitution  whatever  to  this  series,  the  effect  will 
be  simply  to  interchange  the  p  values  among  themselves.  If  in 
particular  the  substitutions  applied  belong  to  the  group  G^  of  <p^ , 
then  the  p  values  will  be  so  interchanged  that  w-^  retains  its  place. 

n\ 
All  the  r  =  ^->{n — 1) !  substitutions  of  G^  therefore  rearrange  only 

the  p  —  1  values  (P2^^zi  •  -  -  ^p-  Since  p  <  n,  there  are  at  the  most, 
only  {p  — 1)!^(^  —  2)!  such  rearrangements.  Consequently  among 
the  r  >  {n  —  1)!  substitutions  of  G^  there  must  be  at  least  two,  <t  and 
T,  which  produce  the  same  rearrangement  of  9^25  9^3)  •••  s^p-  Then 
(TT~^  is  a  substitution  different  from  identity,  which  leaves  all  the 
^p's  unchanged,  that  is,  which  occurs  in  all  the  conjugate  groups 
Gi,  G2,  .  .  .  Gp.  But  if  n  >  4  there  is  no  such  substitution  (Chapter 
III,  Theorem  XIII).     Consequently  p^n. 

§  116.  Passing  to  the  more  general  question  of  the  determina- 
tion of  all  functions  whose  number  of  values  does  not  exceed  a  given 
limit  dependent  on  n,  we  can  dispose  once  for  all  of  the  less  impor- 
tant cases  of  the  intransitive  and  the  non- primitive  groups.  For 
the  purpose  we  have  only  to  employ  the  results  already  obtained  in 
Chapter  IV. 

In  the  case  of  intransitive  groups  we  have  found  for  the  maxi- 
mum orders: 

1)  r=(n  —  1)!.     Symmetric  group  of  n  —  1  elements,     p  =  n. 

U  — 1)1 

2)  r  =  - — -9-^-      Alternating  group  of  n  —  1  elements,     p  =  2n. 

3)  r  =  2 !  (n — 2) ! .    Combination  of  the  symmetric  group  of  ?i — 2 

nin — 1) 
elements  with  that  of  the  two  remaining  elements,     p  = ^ . 

4)  r  =  (n  —  2) ! .  Either  the  combination  of  the  alternating  group 
of  n — 2  elements  with  the  symmetric  group  of  the  two  remaining 
elements;  or  the  symmetric  group  of  n  —  2  elements.  In  both  cases 
p  =  n{n  —  1).     Etc. 

*  L.  Kronecker:  Monalsber.  d.  Berl.  Akad.,  1889,  p.  211. 


THE    NUMBER    OF   VALUES    OF   INTEGRAL   FUNCTIONS.  133 

For  the  non- primitive  groups  we  have 

1)  r  =  2 !  I  -^ !  I  .     Two  systems  of  non-primitivity  containing  each 

■^  elements.  The  group  is  a  combination  of  the  symmetric  groups 
of  both  systems  with  the  two  substitutions  of  the  systems  them- 
selves,     p  —' — — ^-—  .    For  n  =  4,  6, 8,  ...  we  have  />  =  3,  10,  35, . . . 

2)  ?'  =  3!  I  -^!  j  .     Three  systems  of  non-primitivity.    The  group 

is  a  combination  of  the  symmetric  groups  of  the  three  systems  with 

n\ 
the  3 !  substitutions  of  the  systems  themselves,     o  =  — :— .   For 

Q   r  ^    ^^ 

n  =r  6,  9, 12,  .  .  .  we  have  p  =  15,  280,  5770, . . .  VT  V 

3)  r  =  3 1  -^!  I .     As  in  2),  except  that  only  the  alternating  group 

of  the  three  systems  is  employed,     p  =  '  For  w  =  6, 9, 12,. . . 


ii')- 


we  have  p  =  30,  560, 11540, . . . 

The  values  of  p  increase,  as  is  seen,  with  great  rapidity. 

§  117.  In  extension  of  the  results  of  §  113  we  proceed  now  to 
examine  the  primitive  groups  which  contain  substitutions  of  four, 
but  none  of  two  or  of  three  elements. 

Such  a  group  G  must  contain  substitutions  of  one  of  the  two 
types 

The  presence  of  Sg  requires  that  of  Sg^  =  (XaXc)  {xjfic^,  which  belongs 
to  the  former  type.  Disregarding  the  particular  order  in  which  the 
elements  are  numbered,  we  may  therefore  assume  that  the  substitu- 
tion 

^5  "^  \X\X2)  yx^x^) 
occurs  in  the  group  G, 

We  transform  Sg  with  respect  to  all  the  substitutions  of  G  and 
obtain  in  this  way  a  series  of  substitutions  of  the  same  type  which 
connect  Xj ,  a?2  >  ^3  >  ^4  with  all  the  remaining  elements  (Chapter  IV, 
Theorem   XIX).     The   group    G  therefore  includes   substitutions 


134  THEORY    OF    SUBSTITUTIONS. 

similar  to  s^  which  contain  besides  some  of  the  old  elements 
iCi,  iCg,  a?3,  Xi  other  new  elements  x^,Xq,Xt,  .  .  . 

This  can  happen  in  three  different  ways,  according  as  one,  two, 
or  three  of  the  old  elements  are  retained.  Noting  again  that  it  is 
only  the  nature  of  the  connection  of  the  old  elements  with  the  new, 
not  the  order  of  designation  of  the  elements  that  is  of  importance, 
we  recognize  that  there  are  only  five  typical  cases : 

(^1^2)  (^3^5)5       \^iXb)  {XiX^J, 
{XiX^)  {X2X^),       {XiX^}  [X^X^ff 

{x,x,)  {x,x,). 
In  the  first  case,  for  example,  it  is  indifferent  whether  we  take 

{X1X2)  {X^X^},    {X1X2)  {X^Xr,),    {X^X^}  {XiX^)j    [X^Xi)  {X2X^)j 

and  in  the  last  we  may  replace  x^  by  Xi^x^,  or  0^4 ,  etc. 

The  first  and  fifth  cases  are  to  be  rejected,  since  their  presence 
is  at  once  found  to  be  inconsistent  with  the  assumed  character  of 
the  group.     Thus  we  have 

(^1^2)  V'^3'^4)  *  \X1X2)  [X^X^)       =  (X^X^X^jj 
Ll'^i'^a)  {XsX^}  •  {XiX^)  {^XQX^)]    =■  {XiX^X2), 

the  resulting  substitutions  in  each  case  being  inadmissible. 

There  remain  therefore  only  three  cases  to  be  examined,  accord- 
ing as  G  contains,  beside  Sj ,  one  or  the  other  of  the  substitutions 

A.)  {XiXs)  {X2X^), 

B)  {x,x^)  {X2X^), 

C)  (^1^5)  (^3^6)> 

the  first  case  involving  one  new  element,  the  last  two  cases  two 
new  elements  each. 

§  118.     A)     The  primitive  group  G  contains  the  substitutions 

S5  =  (£Cia?2)  (^3^4))     ^4  -^  (^1^3)  (^2^5)1 
and  consequently  also 

t  =  S5S4  =  {x^x^XiX^Xi),     Si  =  tsj;-^  =  (XiX^)  (XtX^). 

Since  t  is  a.  circular  substitution  of  prime  order  5,  it  follows  from 
§83,  Corollary  I,  that  if  n^l,  is  at  least  three-fold  transitive. 
Then  G  must  contain  a  substitution  u,  which  does  not  affect  Xi  but 


THE    NUMBER    OF    VALUES    OF    INTEGRAL   FUNCTIONS.  135 

replaces  Xc^  by  x^  and  x^  by  a?^ .  If  we  transform  Sg  with  respect  to 
this  substitution,  we  obtain 

If  x„  is  contained  among  a?2,  a?3,a?4,  a^g,  then  s'  and  s^  have  only 
one  element  in  common  and  if  Xa  is  contained  among  x^yXg,  ... 
then  s'  and  S5  have  only  one  element  in  common.  Both  alternatives 
therefore  lead  to  the  rejected  fifth  case  of  the  preceding  Section. 

If  n>l^  G  becomes  either  the  alternating  or  the  symmetric 
group.     There  is  in  this  case  no  group  of  the  required  kind. 

For  ti  =  4  it  is  readily  seen  that  there  are  two  types  of  groups 
with  substitutions  of  not  less  than  four  elements,  both  of  which  are 
however  non-primitive. 

Groups  of  the  type  A)  therefore  occur  only  for  n  =  5  or  n  =  6. 

For  n  =  5  we  have  first  the  group  of  order  10, 

If  we  add  to  G^  the  substitution  (t  —  {x^x^x^x^,  we  obtain  a  second 
group  of  order  20 

G,^=\s^,t,ff\  =  \t,<T\, 

The  latter  group  is  that  given  on  p.  39.  G-^  and  G^  exhaust  all  the 
types  for  n  —  h. 

For  n  =  6  we  obtain  a  group  G^  of  the  required  type  by  adding 
to  G2  the  substitution  {x^x^  (^^2^3)-  Since  G^  is  of  order  10,  the 
transitive  group  G^  must  be  at  least  of  order  60  (Theorem  II, 
Chapter  IV).  And  again,  since  {x^x^  {x2X^(t  =  t  =  (ajiajeajgajg^i^s)? 
we  may  write  Gi=  \t,rT,  r\.     We  find  then  that 

rt  =  ffh^,        T(T  =  f(TT\ 

Consequently  from  §  37,  it  follows  [that  G^^  is  of  order  120. 

The  60  substitutions  of  Gi  which  belong  to  the  alternating  group 
from  another  group  of  the  required  type 

G,=  \Gi,  {xiX^)(x2X^)\. 

§  1 19.     B)     In  this  case  G  contains 

and  consequently  the  combination 


136  THEORY    OF     SUBSTITUTIONS. 

These  three  substitutions  are  not  sufficient  to  connect  the  six  ele- 
ments Xi,X2,  .  .  .Xq  transitively,  there  being  no  connection  between 
x^jXi  and  Xi^X2,x-^,XQ.  The  group  must  therefore  (§  83)  contain 
another  substitution  of  the  type  (XaX^)  (XyXs)  which  connects 
Xi,X2yX^,  a?6  with  other  elements.  If  this  substitution  should  con- 
tain three  of  the  elements  Xi,X2,x^,  x^  and  only  one  new  one,  it 
would  have  three  elements  in  common  with  v.  This  would  lead 
either  to  to  the  type  A)  or  to  the  rejected  first  case  of  §  117.  If 
the  new  substitutions  contained  only  one  of  the  new  elements 
XiyX^,x^,  iCfi  and  three  new  ones,  then  we  should  have  the  fifth  case 
of  §  117,  and  this  is  also  to  be  rejected. 

There  remains  only  the  case  where  the  new  substitution  connects 
two  of  the  elements  a?i,  a?2,  a?5,  x^  with  two  others.  It  must  then  be 
of  one  of  the  forms 

(x^x^)  (a?2a?fe),     (xiXa)  (x.Xf,),     (x.Xa)  {x^Xj,\ 
{X2X„)  (XgCCfc),      {x^„)  (x^Xi,),     (x^x„)  (x^Xb). 

Of  these  the  first,  third,  fourth  and  sixth  stand  in  the  relation 
defined  by  C)  to  r,  while  the  first,  second,  fifth  and  sixth  stand  in 
the  same  relation  to  v. 

All  the  groups  B)  therefore  occur  under  either  A)  or  C),  and 
we  may  pass  at  once  to  the  last  case. 

§  120.     C).     In  this  case  the  required  group  contains 

(Tj  =  (XiXz)  (X^X^)       ff2  —  {XxX^  {XzX^^        (T^  =  (Tf-  Vg^Tj  =  (X2X^)  (x^X^). 

We  consider  first  the  case  n  =  6. 

The  elements  iri,ir2,a?5  are  not  yet  connected  with  x^^x^^x^. 
There  must  be  a  connecting  substitution  in  the  group  of  the  type 
(XaXp)  (XyXs),  where  we  may  assume  that  Xa  is  contained  among  the 
the  three  elements  a?i ,  a?2  >  ^s  •  If  ^a  were  X2  or  x^ ,  then  we  should 
obtain,  by  transformation  with  respect  to  o-j  or  (T2,  a  substitution 
(XiXf,)  (XcXa),  SO  that  we  may  assume  a  =  l.  The  possible  cases  are 
then 

(a)     (X1X2)  (x^x^),     (xiX^)  {X2X„,),     (a^x^)  (x^x^)  m  =  3,4, 6. 

(/9)     (x^x^)  (XnX^),  m,  n,p  =  S,  4,  6. 

(r)     (XiX^)  {x2X„)y     (x^x^)  {x^x„),  m,  n  =  3,  4,  6. 


THE    NUMBER    OF    VALUES    OF    INTEGRAL   FUNCTIONS.  137 

The  substitutions  of  the  first  and  second  lines  are  to  be  rejected, 
since  their  products  with  <ri ,  (Tg ,  <^3  lead  to  the  first  case  in  §  117,  or 
directly  to  substitutions  wtth  only  three  elements.  There  remain, 
for  the  different  values  of  m  and  n,  only  the  following  cases : 

iXiX^)  [X^Xi),       [XiX^)  [X^X^), 
{XiX^)  {X^Xq),       (^XiXq)  [X^X^), 

{XiX^}  (ajga^e),     [XiXq)  [x^x^). 

The  second  and  fourth  lines  and  the  third  and  sixth  must  be  rejected, 
since  their  substitutions  have  three  elements  in  common,  the  former 
with  (Ti ,  the  latter  with  <t^.  The  first  line  stands  in  the  same  relation 
to  <Ti  as  the  fifth  to  (T2.  We  need  therefore  consider  only  the  first 
line.  The  product  of  either  of  its  substitutions  by  tJ-j  gives  the 
other.     The  required  group  therefore  contains  beside  <j"i  ,  'J'2  »  ^3  also 

The  group  generated  by  these  four  substitutions 

is  of  order  24.  It  is  non-primitive,  the  systems  of  non-primitivity 
being  x^^x^-,  X2,x^\  and  x^^x^.  It  can  also  be  readily  shown 
that  there  is  no  primitive  group  G^  of  the  required  type  which  con- 
tains G'. 

For  w  =:  6  there  are  only  two  groups  of  the  required  type.  These 
are  the  groups  G^   {r  =  60)  and  G^  (r  =  120)  of  §  118. 

§  121.  If  the  degree  of  the  required  group  is  greater  than  6, 
the  indices  m,  n,  p  of  the  lines  a),  /5),  y)  in  the  preceding  Section 
have  a  correspondingly  larger  range  of  values.  It  is  again  readily 
seen,  however,  that  the  three  cases  a)  are  inadmissible.  But  (/S) 
(r)  both  give  rise  to  groups  which  satisfy  the  required  conditions. 
The  actual  calculation  shows  that  in  every  case  a  proper  combina- 
tion of  the  resulting  substitutions  gives  a  circular  substitution  of 
seven  elements.  Consequently  the  group  G  is  at  least  {n — 6) -fold 
transitive  (§  83). 


138  THEORY    OF    SUBSTITUTIONS. 

If  then  w^9,  G  is  at  least  three-fold  transitive,  and  therefore 
contains  a  substitution  which  does  not  affect  x^ ,  but  interchanges  X2 
and  a?3.  If  we  transform  o-j  with  respect  to  this  substitution,  we 
obtain 

and  since  (t^  has  three  elements  in  common  with  <r,  either  we  have 
the  case  A),  or  a?«  =  x^  and  <j"j  is  equal  to  the  o"^  of  the  preceding 
Section. 

In  the  latter  case  the  subgroup  which  affects  a?i ,  a?2 ,  .  . .  a?^  is 
itself  at  least  simply  transitive*  Combining  with  this  group  the  cir- 
cular substitution  of  seven  elements  we  obtain  a  two-fold  transitive 
group.  Consequently  (§  84)  G  is  at  least  (n —  5) -fold,  and  for 
w>^9,  at  least  4-fold  transitive.  G  contains  therefore  the  sub- 
stitutions 

T  =  (xi)  (x^x^)  {x^x^ .  .  .), 

r-'(T,T  =  {x,Xs)(x2X,), 
so  that  we  return  in  every  case  to  the  type  A).     For  n>_9  there  is 
therefore  no  group  of  the  required  type. 

Theorem  III.  If  the  degree  of  a  group,  which  contains 
substitutions  of  four,  but  none  of  three  or  of  two  elements,  exceeds 
8,  the  group  is  either  intransitive  or  non-primitive. 

Combining  this  result  with  those  of  §  113  and  §  116,  we  have 

Theorem  IV.  If  the  number  p  of  the  values  of  a  function 
is  not  greater  than  in{n  —  1),  then  if  n>8,  either  1)  p  =  in(n  —  1), 
and  the  function  is  symmetric  in  n  —  2  elements  on  the  one  hand 
and  in  the  two  remaining  elements  on  the  other ^  or  2)  p  =  2n,  and 
the  function  is  alternating  in  n — 1  elements,  or  d)p  =  n,and 
the  function  is  symmetric  in  n  —  1  elements,  or  4:)  p  =  lor  2,  and  the 
function  is  symmetric  or  alternating  in  all  the  n  elements.* 

§  122.     We  insert  here  a  lemma  which  we  shall  need  in  the 

proof  of  a  more  general  theorem,  f 

From   §  83,  Corollary  II,  a  primitive  group,  which  does  not 

include  the  alternating  group,  cannot  contain  a  circular  substitution 

« 

•Cauchy :  Journ.  de  I'Ecole  Polytech.  X  Cahier;  Bertrand:  Ibid.  XXX  Cahier;  Abel: 
Oeuvres  completes  I,  pp.  13-21;  J.  A.  Serret:  Journ.  del'Ecdle  Polytech.  XXXII  Cahier; 
C.  Jordan:  Trait6  etc.,  pp.  67-75. 

t  C.  Jordan :  Trait6  etc..  p .  664.    Note  C. 


THE    NUMBER   OF    VALUES    OF    INTEGRAL    FUNCTIONS.  139 

of  a  prime  degree  less  than  -^ .     If   p   is  any  prime  number  less 

o 
o 

than  -K- ,  and  if  p^  is  the  highest  power  of  p  which  is  contained  in 

o 

n!,  then  the  order  of  a  primitive  group  G  is  not  divisible  byp^. 

For  otherwise  G  would  contain  a  subgroup  which  would  be  similar 

to  the   group  K  of  degree  n  and  order  p^  (§  39).     But  the  latter 

group  by  construction  contains  a  circular  substitution  of  degree  p, 

n\ 
and  the  same  must  therefore  be  true  of  G.     Consequently  p  =  ^ 

must  contain  the  factor  p  at  least  once. 

What  has  been  proven  for  p  is  true  of  any  prime  number  less 

2n 
than  -^  and  consequently  for  their  product.     We  have  then 

o 

Theorem  V,  If  the  group  of  a  function  with  more  than 
two  values  is  primitive,  the  number  of  values  of  the  function  is  a 
multiple  of  the  product  of  all  the  prime  numbers  which  are  less 

than  -^. 
o 

§  123.     By  the  aid  of  this  result  we  can  prove  the  following 

Theorem  VI.  If  k  is  any  constant  number,  a  function  of 
n  elements  which  is  symmetric  or  alternating  with  respect  to  n  —  k 
of  them  ha^  fewer  values  than  those  functions  which  have  not  this 
property.  For  small  values  of  n  exceptions  occur,  but  if  n  exceeds 
a  certain  limit  dependent  on  k,  the  theorem  is  rigidly  true.* 

If  (f  is  an  alternating  function  with  respect  to  n — k  elements, 
the  order  of  the  corresponding  group  is  a  multiple  of  ^{n — k)\,  and 
the  number  of  values  of  the  function  is  therefore  at  the  most 

A)  2n{n—l)  (n -  2) . . .  (n— ^4- 1). 

If  ^  is  a  function  which  is  neither  symmetric  nor  alternating  in 
n — k  elements,  it  may  be  transitive  with  respect  to  n — A;  or  more 
elements.  But  in  the  last  case  (p  must  not  be  symmetric  or  alterna- 
ting in  the  transitively  connected  elements. 

We  proceed  to  determine  for  both  cases  a  minimum  number  of 

*C.  Jordan:  Traits  etc.,  p.  67. 


140  THEORY    OF    SUBSTITUTIONS. 

values  of  0,  and  to  show  that  if  n  is  sufficiently  large,  this  mini 
mum  is  greater  than  the  maximum  number  of  values  A)  of  <p. 

§  124.  Suppose  at  first  that  ^  is  transitive  in  less  than  n — k 
elements.     Then  the  order  of  the  corresponding  group  is  a  divisor 

of 

X^\  X^\  iig!  .  .  .  where  X^-[-  X,^-{-  X^-\-  .  .  .  =  n     (^a  <  "^ — k). 

This  product  is  a  maximum  when  one  of  the  A's  is  as  large  as  pos- 
sible, i.  e.,  equal  to  n — k  —  1,  and  a  second  A  is  then  also  as  large 
as  possible,  i.  e.,  equal  to  k-{-l.  It  is  further  necessary  that 
k-\-l>n — k  i.  e.,  n>2k-{-l.  The  maximum  order  of  the  group 
is  consequently 

(n— A;— 1)!  (A;  +  l!, 

and  the  minimum  number  of  values  of  <l>  is 

n ' n{n—l){n  —  2)..,{n  —  k) 

^  {n—k—l)\{k-j-l\)~        1.2.3.  ...(&+ 1) 

It  appears  at  once  that  the  minimum  B)  exceeds  the  maximum  A), 
as  soon  as 

n>k-\-2{k+l)l 

This  is  therefore  the  limit  above  which,  in  the  first  case,  the  theo- 
orem  admits  of  no  exception. 

§  125.  In  the  second  case  (p  is  transitive  in  n  —  x  elements 
(z>_/f),  but  it  is  neither  alternating  nor  symmetric  in  these  ele- 
ments. The  group  G  of  (J>  is  intransitive,  and  its  substitutions 
are  therefore  products  each  of  two  others,  of  which  the  one  set 
^1)  ^2j  •  • '  connect  transitively  only  the  elements  Xi,X2,  .  .  .  x„_^, 
while  the  other  set  t^^t^^  .  .  .  connect  only  the  remaining  elements 
x„  _  It  ^_  1 , .  . .  x„. 

The  substitutions  of  the  group  G  of  </'  have,  then,  the  product 
forms 

where,  however,  one  and  the  same  (t  may  occur  in  combination  with 
different  r's.  It  is  easily  seen  that  all  the  (t's  occur  the  same  num- 
ber of  times,  so  that  the  order  of  the  group  G  is  a  multiple  of  that 
of  the  group  2  =  [<ri ,  trg ,  . . .  ]. 

We  will  show  that  S  is  neither  alternating  nor  symmetric;  oth- 


THE    NUMBER    OF    VALUES    OF    INTEGRAL    FUNCTIONS.  141 

erwise  G  would  be  alternating  or  symmetric  in  n — z  elements, 
which  is  contrary  to  assumption.  If  the  group  2  were  alternating, 
it  would  be  of  order  J(n  —  z)!.  This  exceeds  the  maximum  num- 
ber z!  of  the  order  of  the  group  r=  [r^, -3,  . .  .  ]  of  z  elements, 
as  soon  as  w  >  2k.  Consequently  G  contains  substitutions  (r^j-^ , 
fl-|3r|3,  in  which  t^^t^  but  'J'a4"^^j  ^"^^  therefore  substitutions 
ffaJa  ((Tprp)~^  =  (TaTp~^  which  affect  only  the  elements  a?j,  a?2,  .  .  .oc^-k 
of  the  first  set.  The  entire  complex  of  these  substitutions  forms  a 
self- conjugate  subgroup  H  of  G.  This  subgroup  is  unchanged  by 
transformation  with  respect  to  either  G  or  S,  since  r^,  r^,  ...  have 
no  effect  whatever  on  the  substitutions  of  H.  H  is  therefore  a  self - 
conjugate  subgroup  of  the  alternating  group  2 ,  and  must  accord- 
ingly coincide  with  2  (§  92).  if  =  S  is  therefore  a  subgroup  of  6r, 
and  4'  would,  contrary  to  assumption,  be  alternating  in  w  —  x  ele- 
ments. 

§  126.  The  maximum  order  of  the  group  G  is  therefore  equal 
to  the  product  of  z!  by  the  maximum  order  of  a  non- alternating 
transitive  group  of  n — /.  elements.  We  denote  the  latter  order 
by  R{n  —  z).     Then  the  minimum  number  of  values  of  4'  is 

.  n\  _  {n — x)!    n{n  —  l).,.{n — x -|- 1) 

^  AR{n—x)~  R{n  —  y)'  x\  ' 

We  have  now  still  to  determine  R{n  —  x),  the  maximum  order  of 
a  non- alternating  transitive  group  of  n  —  x  elements,  or  -^. -A- , 

the  minimum  number  of  values  of  a  non- alternating  transitive 
function  oi  n  —  x  elements. 

If  this  function  is  non-primitive  in  the  n — x  elements,  it  follows 
that  the  minimum  number  of  values  is 


(„_.)(„_,^i),...('i_j'+i) 


Substituting  this  value  in  C)  we  obtain  for  the  minimum  number  of 
values  of  4' 

n{n-l).  .  ,  (n-x  +  l)  (n-x)  .  .  .  fl-^+l) 


142  THEORY    OF    SUBSTITUTIONS. 

We  compare  this  number  with  the  maximum  number  A)  and 
examine  whether,  above  a  certain  limit  for  n,  C\)  becomes  greater 
than  A),  i.  e,,  whether 

>4x!  !!LZ^in{n—l).  ,  .(n— x  +  1). 

For  sufficiently  large  n  we  have  I  — ~  +  l)  <*^ — k-{-l.  We  have 
therefore  to  prove  that 


n — X 


This  is  shown  at  once,  if  we  write  the  right  hand  member  in  the 
form 

(4.![,_.]o([!^^]['i^_l]...[.-.  +  l]). 

For  the  first  factor  is  constant  as  n  increases,  and  the  ratio  of  the 
left  hand  member  to  the  second  parenthesis  has  for  its  limit 

n-\-K 

2—-'-. 

§  127.  Finally,  if  the  function  <p  of  the  n — x  elements  is  prim- 
itive, we  recur  to  the  lemma  of  §  122.  From  this  it  follows  that 
the  minimum  number  of  values  of  4'  is  the  product  of  all  the  prime 
numbers  less  than  §  (n — x).     We  will  denote  this  product  by 

Introducing  it  in  C),  we  have 

We  have  then  to  show  that,  for  sufficiently  large  values  of  n,  the 
value  A)  is  less  than  Ca),  *.  e.,  that 

^j^2(n— x)j  ^  2x\{n—x)  (n—x  —  l) . . .  {n~k-{-l). 

The  right  hand  member  of  this  inequality  will  be  greatly  increased 
if  we  replace  every  n — x  —  a  by  the  first  factor  n — x.  There  are 
k — X  factors  of  the  form  n — x  —  a.     These  will  be  replaced  by 


THE    NUMBER    OF    VALUES    OF    INTEGRAL    FUNCTIONS.  143 

(n — x)*^-*.     If  we  write  then  V  = —^-^ — -,  we  have  only  to  prove 
that  for  sufficiently  large  v, 

or 

p(.) 


.Je- 


>[2(|f-«.!]. 


This  can  be  shown  inductively  by  actual  calculation,  or  by  the 
employment  of  the  theorem  of  Tchebichef ,  that  if  j^  >  3,  there  is 
always  a  prime  number  between  v  and  2v  —  2. 

For  we  have  from  this  theorem 

P(2v)>vP(v), 

P(2v)  P(v)       V 


> 


Now  whatever  value  the  first  quotient  on  the  right  may  have,  we 
can  always  take  t  so  great  that  the  left  hand  member  of 

P(2-v)      P(.)  f    V   Y 

increases  without  limit,  if  only  v  is  taken  greater  than  2*'"".     The 
proof  of  the  theorem  is  now  complete. 

The  limits  here  obtained  are  obviously  far  too  high.  In  every 
special  case  it  is  possible  to  diminish  them.  As  we  have,  however, 
already  treated  the  special  cases  as  far  as  ^o  =  ^n(n — 1),  it  does  not 
seem  necessary,  from  the  present  point  of  view, -to  carry  these  inves- 
tigations further. 


CHAPTER  VII. 


CERTAIN  SPECIAL  CLASSES  OF  GROUPS. 

§  128.  We  recur  now  to  the  results  obtained  in  §  48,  and 
deduce  from  these  certain  further  important  conclusions.* 

Suppose  that  a  group  G  is  of  order  r=p'^m,  where  p  is  a  prime 
number  and  m  is  prime  to  p.  We  have  seen  that  G  contains  a  sub- 
group H  of  order  p".  Let  J  be  the  greatest  subgroup  of  G  which 
is  commutative  with  H.  J  contains  H,  and  the  order  of  J  is  there- 
fore pH,  where  i  is  a  divisor  of  m  and  is  consequently  prime  to  p. 

Excepting  the  substitutions  of  H,  J  contains  no  substitution  of 
an  order  p^.  For  if  such  a  substitution  were  present,  its  powers 
would  form  a  group  L  of  order  p^.  But  if  in  A)  of  §  48  we  take 
for  Gii  Hi,  Ki  the  present  groups  «/,  X,  jK",  then  since  <Ty~~^H(Ty  =  Hy 
we  should  have 

pH  _p^  .p^  . 
'p^~d^d^'^  '  " 

The  left  member  of  this  equation  is  not  divisible  by  p.  Conse- 
quently we  must  have  in  at  least  one  case  dy  =  jp^,  that  is,  L  is  con- 
tained in  ffy^^Hffy  =  H. 

Again,  every  subgroup  M  of  order  p*  which  occurs  in  G  is 
obtained  by  transformation  of  H.  For  if  we  replace  (tj  ,  H^,  K^  of 
A)  §  48  by  G,  H,  M,  we  obtain 

and  for  the  same  reason  as  before  dy^p'^  in  at  least  one  case,  and 
therefore  M=(Ty~  ^H^^ .  Since  if,  as  a  self- conjugate  subgroup  of  Jy 
is  transformed  into  itself  by  the  p'^i  substitutions  of  J,  it  follows  that 
there  are  always  exactly  pH  substitutions  of  G  which  transform  H 

into  any  one  of  its  conjugates.     There  are  therefore  -^  of  the  latter. 

t 

*  L.  Sylow :  Math.  Ann.  V.  584-94. 


CERTAIN    SPECIAL    CLASSES    OF    GROUPS.  145 

Finally,  if  we  replace  G,  H^ ,  K^  of  A)  §  48  by  G,  J,  if,  we 
have 

r  _  p'^m  __  pH     pH 

Since  jffj  is  contained  in  J^ ,  we  must  have  d^  =p%  and  since  J  con- 
tains no  other  substitutions  of  order  p^,  no  other  d  can  be  equal  to 
p*.     It  follows  that 

r  =  pH  (kp  +  1),     m  =  i(kp  -j-  1) . 

The  group  H  has  therefore  kp-{-l  conjugates  with  respect  to  G, 
We  have  then  the  following 

Theorem  I.     //  the  order  r  of  a  group  G  is  divisible  by  p"^ 

but  by  no  higher  power  of  the  prime  number  p,  and  if  H  is  one  of 

the  subgroups  of  order  p"  contained  in  G,  and  J  of  order  pH  the 

largest  subgroup  of  G  which  is  commutative  with  H,  then  the  order 

of  G  is     ,  . 

r=pH{kp -{-!). 

Every  subgroup  of  order  p"^  contained  in  G  is  conjugate  to  H.  Of 
these  conjugate  groups  there  are  kp-\-l,  and  every  one  of  them 
can  be  obtained  from  H  by  pH  different  transformations. 

§  129.  In  the  discussion  of  isomorphism  we  have  met  with  tran- 
sitive groups  whose  degree  and  order  are  equal.  In  the  following 
Sections  we  shall  designate  such  groups  as  the  groups  9.. 

If  we  regard  all  simply  isomorphic  transitive  groups,  for  which 
therefore  the  orders  r  are  all  equal,  as  forming  a  class,  then  every 
such  class  contains  one  and  only  one  type  of  a  group  ii  (§  98).  The 
construction  of  all  the  groups  ii  of  degree  and  order  r  therefore 
furnishes  representatives  of  all  the  classes  belonging  to  r,  together 
with  the  number  of  these  classes.  The  construction  of  these  typi- 
cal groups  is  of  especial  importance,  because  isomorphic  groups 
have  the  same  factors  of  composition,  and  the  latter  play  an  impor- 
tant part  in  the  algebraic  solution  of  equations. 

One  type  can  be  established  at  once,  in  its  full  generality.     This 
type  is  formed  by  the  powers  of  a  circular  substitution.     A  group  ii 
of  this  type  is  called  a  cyclical  group,  and  every  function  of  n  ele- 
ments which  belongs  to  a  cyclical  group  is  called  a  cyclical  function. 
10 


146 


THEOBY    OF    SUBSTITUTIONS. 


We  limit  ourselves  to  the  consideration  of  cyclical  groups  of 
prime  degree  p.  If  s  =  (x^  x^ .  . .  x^),  and  if  w  is  any  primitive  p^^ 
root  of  unity,  then 

is  a  cyclical  function  belonging  to  the  group  G  =  [1,  s,  s%  .  .  .  s^~n. 
For  <p  is  converted  by  s*  into 

so  that  <p  is  unchanged  by  the  substitutions  of  G. 

Moreover,  if  for  any  substitution  t,  which  converts  every  x^  into 
Xf^,  we  have  <pt  =  <p^^  and  consequently 

then,  if  the  a^'s  are  independent  elements,  it  follows  that 


^Xt^  =  <^     ^<^*'      -*y. 


Consequently  ty  =  Y-{-  iS,  that  is,  the  substitution  t  replaces  a^i ,  a?2. . . , 
by  0*1^/3, 0*2+ )3)  •••;  ^  is  therefore  contained  among  the  powers  of  s, 
and  <p  belongs  to  the  group  G.  It  is  obvious  that  for  r  =p,  the 
group  G  furnishes  the  only  possible  type  ^. 

§  130.  We  proceed  next  to  determine  all  types  of  groups  ^  of 
degree  and  order  pq,  where  p  and  q  are  prime  numbers,  which  for 
the  present  we  will  assume  to  be  unequal,  p  being  the  greater. 

1)  One  type,  that  of  the  cyclical  group,  is  already  known  to  us. 
It  is  characterized  by  the  occurrence  of  a  substitution  of  order  pq'. 

'2)  If  there  are  any  other  types,  none  of  them  can  contain  a 
substitution  of  order  pq,  since  this  would  lead  at  once  to  1).  The 
only  possible  orders  are  therefore  p,  Q',  1.  A  substitution  s  of  order 
p  is  certainly  present.  This  and  its  powers  form  within  ^  a  sub- 
group H  of  order  p.  If  ^  contains  any  further  subgroups  H'  of 
order  p,  their  number  must  be  xp  -j- 1,  where  x  >  0.  These  sub- 
groups would  have  only  the  identical  substitution  in  common. 
They  would  therefore  contain  in  all 

(p-l)(px-hl)  +  l=p[(p-l)x  +  l]>pg 
substitutions.     This  being  impossible,  we  must  have  x  =  0. 


CERTAIN    SPECIAL    CLASSES    OF    GROUPS.  147 

The  subgroup  H  contains  only  p  substitutions;  the  rest  are  all 
of  order  q.     Their  number  is 

pq—p={q  —  l)p. 
There  are  therefore  p  subgroups  of  order  g,  and  consequently  from 
Theorem  I  we  must  have 

that  is,  q  must  be  a  divisor  of  p  —  1.  Only  in  this  case  can  there 
he  any  new  type  ^. 

3)  The  group  H  ia  a.  self-conjugate  subgroup  of  ^.  Conse- 
quently every  substitution  t  of  order  q  must  transform  the  substitu- 
tion s  of  H  into  s",  where  a  might  also  be  equal  to  1.     We  write 

(where  the  upper  indices  are  merely  indices,  not  exponents).  Then 
no  cycle  of  t  can  contain  two  elements  with  the  same  upper  index. 
For  otherwise  in  some  power  of  t  one  of  these  elements  would 
follow  the  other,  and  if  this  power  of  t  were  multiplied  by  a  proper 
power  of  s,  one  of  the  elements  would  be  removed. 

With  a  proper  choice  of  notation,  we  may  therefore  take  for  one 
cycle  of  t 

\      1        1        1     •    •    •  *^1    /• 

It  follows  then  from 

t-^st  =  s'' 

that  t  replaces  x^^  by  a?„  +  i*  +  ',  x^^  by  X2a  +  i^'^\  .  .  .  Xa+i^  by 
a?a«  +  i^^\  .  .  •  so  that  we  have 

t  =  yXi  Xi     .   .  .  Xi  )  .  ,  .  (iCft  ^  I  Xaa  -\- 1  ^aa^  +  1     •   •  •  ^aa9  —  1+1     ...).•• 

If  now  the  latter  cycle  is  to  close  exactly  with  the  element  Xaa^-^+i% 
we  must  have 

aa^-{-l^a-\-l,     a^^l     (mod.  p). 

The  solution  a:=^l  is  to  be  rejected,  for  in  this  case  we  should  have 

T \P^i  «^l     •   •   •  X-^    )  ( M?2  "^2     •   •   •  '^2  )  '   '   '   \p^p  '^p     «   •   •  Xp    ), 

St  ^^^  (Xi  X2     .  .  .  Xq  Xq  _[.  I  Xq  _^ 2     •   •   •  7   •   •   •  i 

SO  that  the  latter  substitution  would  contain  a  cycle  of  more  than  q 
elements,  without  being  a  power  of  s. 


148  THEUJtv/    OF    SUBSTITUTIONS. 

It  follows  then  from  the  congruence  a*^l  (mod.  p)  that  q*  is  a 
divisor  of  p  —  1,  as  we  have  already  shown;  further  that  a,  belong- 
ing to  the  exponent  g,  has  q — 1  values  a^,a2,  .  .  .  a^-n  finally  that 
all  these  values  are  congruent  (mod.  p)  to  the  powers  of  any  one 
among  them.     From  t^^st  —  s"  follows 

so  that,  if  s  is  transformed  by  t  into  any  one  of  the  powers  s*a,  there 
are  also  substitutions  in  ^  which  transform  s  into  s"i,s"2,  .  .  .  s"-?-'. 
Accordingly  the  particular  choice  of  a\  has  no  influence  on  the 
resulting  group,  so  that  if  there  is  any  type  ^  generated  by  substi- 
tutions s  and  t^  there  is  only  one. 

The  group  formed  by  the  powers  of  t  being  commutative  with 
that  formed  by  the  powers  of  s,  the  combination  of  these  two  sub- 
stitutions gives  rise  to  a  group  exactly  of  order  pg.  The  remaining 
pq — p  —  q-\-l  substitutions  of  the  group  are  the  first  ^  —  1  powers 
of  the  p  —  1  substitutions  conjugate  to  t 

s-^+Hs^-'  =  (cc^VV  .  •  .  ^^') . .  •     ('5  =  2.  3,  ...p). 
If  p  and  q  are  unequal,  we  have  therefore  only  one  7ieio  type  9., 

§  131.  Finally  we  determine  all  types  of  groups  il  of  degree 
and  order  pi 

1)  The  cyclical  type,  characterized  by  the  presence  of  a  substi- 
tution of  order  pi\  is  already  known. 

2)  If  there  are  other  types,  none  of  them  can  contain  a  sub- 
stitution of  order  pi  There  are  therefore  in  every  case  p^ — 1  sub- 
stitutions of  order  p  and  one  of  order  1.  If  s  is  any  substitution 
of  ^,  and  t  any  other,  not  a  power  of  s,  then  9  is  fully  determined 
by  s  and  t.     For  all  the  products 

sH''     (a,6  =  0,l,2,  ...p  — 1) 

are  different,  and  therefore 

i2=z[8"f]     (a,5  =  0,l,2,  ...p  — 1). 
We  must  have  therefore 

ts  =  s«i  ^^1 ,     fs  =  s^^  i%...i ''  -  's  ^  6-«^' " '  t^J'  - 1 . 
If  now  two  of  the  exponents  d  are  equal,  it  follows  from 


CERTAIN    SPECIAL    CLASSES    OF    GROUPS.  149 

rs=:sH%     fs  =  sH"     (a-^b,     sis') 
that 

(rs)-\fs)  =  s-H's  ^  (sH')-\sH")  ■:=  fy. 

Since  for  t"  we  may  write  t,  it  therefore  appears  that  ^  contains  a 
substitution  t  which  is  transformed  by  s  into  one  of  its  powers  P. 

The  same  result  holds,  if  all  the  exponents  S  are  different.  For 
one  of  them  is  then  equal  to  1,  since  none  of  them  can  be  0,  and 
from  Ts  =:  st^  follows  s~H"s  =  t^. 

3)     There  is  therefore  always  a  substitution 

T  —  [Xi  X^    .  .  .  Xp  J  [X^  X2    .  .  .  Xp  j  .  .  .  (^1  X2    ...  Xp  ) 

which  is  transformed  by  s  into  a  power  of  itself  t"^.  As  in  the  pre 
ceding  Section,  we  may  take  for  one  cycle  of  s 

(x^^x^^ .  .  .  X^P). 

Then  from  s'Hs^^r  follows 

S  =  (Xi^X,^  .  .  .  Xf)  {X2^Xa  +  i^Xa2  +  i^  .  .  .  XaP-^+Z'  ...)... 

If  the  second  cycle  is  to  close  after  exactly  p  elements,  we  must 
have 

a^  +  l  =  2,     a^=l     (mod  p). 
This  is  possible  only  if  a  =  1.     Accordingly 

S  —  [Xi  Xi    .  .  .  Xi  )  yX2  X2    ,  .  .  X2  )  .  .  .  \Xp  Xp    .  ,  .  p  j. 

The  p-\-l  substitutions 

are  all  different  and  no  one  of  them  is  a  power  of  any  other  one. 
Their  first  p  —  1  powers  together  with  the  identical  substitution 
form  the  group  i2. 

Summarizing  the  preceding  results  we  have 

Theorem  II.  There  are  three  types  of  groups  ^,  for  which 
the  degree  and  order  are  equal  to  the  product  of  two  prime  num- 
bers: 1)  The  cyclical  type,  2)  one  type  of  order  pq  (p>q)^ 
3)  one  type  of  order  pi  The  first  and  third  types  are  always  pres- 
ent; the  second  occurs  only  when  q  is  a  divisor  of  p — 1. 

§  132.  We  consider  now  another  category  of  groups,  character- 
ized by  the  property  that  their  substitutions  leave  no  element,  or 


150  THEORY    OF   SUBSTITUTIONS. 

only  one  element,  or  all  the  elements  unchanged.  The  degree  of 
the  groups  we  assume  to  be  a  prime  number  p. 

Every  substitution  of  such  a  group  is  regular,  i.  e.,  is  composed 
of  equal  cycles.  For  otherwise  in  a  proper  power  of  the  substitu- 
tion, different  from  the  identity,  two  or  more  of  the  elements  would 
be  removed. 

The  substitutions  which  affect  all  the  elements  are  cyclical,  for  p 
is  a  prime  number.  From  this  it  follows  that  the  groups  are  tran- 
sitive, and  again,  from  Theorem  IX,  Chapter  IV,  that  the  number 
of  substitutions  which  affect  all  the  elements  is  p  —  1.  We  may 
therefore  assume  that 

and  its  first  p  — 1  powers  are  the  only  substitutions  of  p  elements 
which  occur  in  the  required  group. 

The  problem  then  reduces  to  the  determination  of  those  substi- 
tutions which  affect  exactly  p  —  1  elements.  If  t  is  any  one  of  these, 
then  t~^st,  being  similar  to  s,  and  therefore  affecting  all  the  ele- 
ments, must  be  a  power  of  s 

where  every  index  is  to  be  replaced  by  its  least  positive  remainder 
(mod  p).  Since  it  is  merely  a  matter  of  notation  which  element  is 
not  affected  by  t,  we  may  assume  that  x^  is  the  unaffected  element. 
It  follows  that 

If  now  gr  is  a  primitive  root  (mod.  p),  then  all  the  remainders 
(mod.  p)  of  the  first  p — 1  powers  of  g 

G)  g\cf,g\...g^-\g^-'=l     (mod.  p) 

are  different,  and  we  may  therefore  put 

m^gf^     (mod.  p). 

We  will  denote  the  corresponding  thyt^.     It  appears  then  that  t^ 

p 1 

consists  of  At  cycles  of  — elements  each.     For  every  cycle  of  t^ 

closes  as  soon  as 


CERTAIN    SPECIAL    CLASSES    OF    GROUPS.  151 

am*+l  =  a+l, 

^^      ^       (mod.  p) 

and  this  first  happens  when  z  — . 

If  there  is  any  further  substitution  t^  which  leaves  x^  unchanged 
and  which  replaces  every  a?«  +  i  by  a?„/+i,  then  iy^tv^  replaces  every 
a?a  +  i  by  a?„^aM  +  /3v^i.  If  now  we  take  «  and /5  so  that  aix-\-^v  is 
congruent  (mod.  p)  to  the  smallest  common  divisor  o)  of  /^  and  \>y 
we  have  in 

^Oi  —   *'|U,    f^v 

a  substitution  of  the  group,  of  which  both  t^  and  f^  are  powers. 

Proceeding  in  this  way,  we  can  express  all  the  substitutions  which 

leave   x^  unchanged   as  powers  of  a  single  one  among  them  t^j 

where  g*^  is  the  lowest  power  of  g  to  which  a  substitution  t  of  the 

group  corresponds. 

»— 1 
The  group  is  determined  by  s  and  t^^.    Since  t^^  is  of  order , 

it  follows  from  Theorem  II,  Chapter  IV,  that  the  group  contains  in 

p(p  —  1) 
all  ^ substitutions.     <t  may  be  taken  arbitrarily  among  the 

divisors  of  p  —  1. 

§  133.  To  obtain  a  function  belonging  to  the  group  just  con- 
sidered, we  start  with  the  cyclical  function  belonging  to  s 

^1 ={xi-\-(ox2-^-  to^xs  +  • .  •  +  <^^~'^x^y, 

where  (o  is  any  primitive  p*^  root  of  unity.  Applying  to  ^^  the 
successive  powers  of  ^^j  we  obtain 

<p3'={Xl-\-^Xg^-'r  +  i  +  Oj'x2g2<r^,-{-  .  .  .  -\-iO^-'x<^^_,)^2<T  +  ,y, 

The  powers  of  s,  forming  a  self-conjugate  subgroup  of  the  given 
group,  leave  all  the  ^'s  unchanged.  The  powers  of  t^,  and  conse- 
quently all  the  substitutions  s'^tj'  of  the  group,  merely  permute  the 
0's  among  themselves.     Every  symmetric  function  of 

<Pu<p2i   '"4'p-l 


152  THEORY    OF     SUBSTITUTIONS. 

is  therefore  unchanged  by  every  substitution  of  the  group.  Ac- 
cordingly if  il'  is  any  arbitrary  quantity,  the  function 

^;  =  {4'-4',)  {4'— 4',) . . .  {4'-4',_^ 

is  unchanged  by  all  these  substitutions  and  by  no  others.  ^„  there- 
fore belongs  to  the  given  group. 

§  134.  If,  in  particular,  we  take  (^  =  \,  the  order  of  the  group 
becomes  p{p —  1).     The  substitution  t^  is  then  of  the  form 

containing  only  one  cycle.  The  group  is  therefore  two-fold  transi- 
tive (Theorem  XIII,  Chapter  IV).  A  group  of  this  type  is  called  a 
metacycUc  group,  and  the  corresponding  •  function  ^^\  a  metacyclic 

function. 

jylt) —  1^ 
If  0-  =  2,  the  order  of  the  group  is  ^^^ ,  and  t„  is  of  the 

form 

^2  =  (a?2^gr2+l^sr4_|_i   .   .    .)  {X^  j^yXa^  j^iXagi  J^i   .   .   .) 

The  indices  in  the  first  cycle,  each  diminished  by  1,  are  the 
quadratic  remainders  (mod.  p) ;  a  is  any  quadratic  non-remainder 
(mod.  p).  The  group  is  in  this  case  called  half -metacyclic  group 
and  the  corresponding  function  W^.  a  half -metacyclic  function.* 

§  135.  We  can  define  all  substitutions  of  the  groups  i^  of 
prime  degree  p,  as  well  as  those  of  the  two  preceding  Sections,  in  a 
simple  way  by  expressing  merely  the  changes  which  occur  in  the 
indices  of  the  elements  a?,,  a?2)  •  •  •  ^w  Thus  the  substitutions  of 
Q  are  defined  by 

Sa=\z  z-\-a\      (mod.  p).     (a  =  0,1,2,  .  ..p  — 1), 

The  symbol  here  introduced  is  to  be  understood  as  indicating  that 
in  the  substitution  Sa.  every  element  Xg  is  replaced  by  x^j^^^  and  for 
js;  +  a  its  least  not  negative  remainder  (mod.  p)  is  to  be  taken. 

The  groups  of  the  preceding  Section  contain  then,  in  the  first 
place,  all  the  substitutions  s^,  and  beside  these,  if  we  suppose 
every  index  to  be  diminished  by  one,  also  those  substitutions  for 
which  every  index  is  multiplied  by  the  same  factor,  that  is,  for 
>i  =  0,l,2,...ff-l . 

*  L.  Kronecker;  cf.  F.  Klein:  Math.  Ann.  XV,  258. 


CERTAIN  SPECIAL  CLASSES  OF  GROUPS.  153 

^p=\z  lSz\     (mo(J.p)     (/5  =  1,2,3,  ...p-1). 
The  symbol 

t=\z  fiz-\-a\     (modp)     (6c  =  0,l,  ...p  — 1;     /9=  1,  2,  .  .  .p  — 1) 
includes  all  the  substitutions  Sa,  f^p,  and  their  combinations.     Since 

\z     I3z  +  a\.\z     l^^z-\-a,\  =  \z     l3,3,z-{-a,l3-\-a\, 

it  follows  that  the  substitutions  t  form  a  group  of  degree  p  and  of 
order  jp(p  —  1).     This  group  therefore  coincides  with  that  of  §  134. 
If  we  prescribe  that  the  /?'s  shall  take  only  the  values 

the  products  /?/?i  belong  to  the  same  series,  and  we  obtain  the  group 

vip — 1) 
of  degree  p  and  of  order  — considered  above. 

§  136.  The  consideration  of  the  fractional  linear  substitutions 
(mod.  p)  leads  to  groups  of  degree  p  +  1  and  of  order  {p-\-V)p 
(p  —  1).     These  substitutions  are  of  the  form 

az-{-  [3 


(mod.  p), 


yz-^ 

where  z  is  to  take  the  values  0, 1,  2,  .  .  .p  —  1,  oo  ,  the  elements  of 

the  group  being  accordingly  Xq,XisX2,  .  .  .  x^_i,x^.     The  values 

«j  /^j  ?')  ^1   determine  a  single  substitution  u,  but  it   may  happen 

that  one  and  the  same  u  results  from  different  systems  a,  /?,  y,  d. 

To  avoid,  or  at  least  to  limit  this  possibility,  we  make  use  of  the 

determinant  of  u 

D)  ad  —  ftr. 

According  as  this  is  a  quadratic  remainder  or  non- remainder,  we 

divide  numerator  and  denominator  of  — ^|-^  by  A^ad — ^y  or  by 

\/l3y — ad.     For  the  new  coefficients  we  have  then 

D')  ad  —  i3y=±i      (mod.  p). 

If    now   for   two   different   systems   of   coefficients    a,  /5,  y,  S    and 
^i»/^i)rj^i  the  relation 

az-\-l3___a,z-^r 


yz-i-d      y,z  + 


(mod.  p) 


154  THEORY    OF    SUBSTIT  0TI0N8. 

were  possiblGj  it  would  follow  from  the  comparison  of  the  coeffi- 
cients of  ^^  z^,  and  z^,  with  the  aid  of  D',  that  if  a,  a',  /5,  /3',  . , .  are 
real, 

a  13         r  8  I  ad  — fir  .       /        -.       x 

If,  therefore,  we  restrict  the  range  of  the  values  of  a,  /?,  /',  d  to 
0,1,2,  ...  p  —  1,  there  are  always  two  and  only  two  different  sys- 
tems of  coefficients  which  give  the  same  substitution  u. 

With  D')  it  is  assumed  that  ad — i3y  is  different  from  0.     This 
restriction  is  necessary,  for  the  symbol  u  can  represent  a  substitu- 
tion only  if  different  initial  values  of  z  give  rise  to  different  jSnal 
values  of  2,  i.  e.,  if  the  congruence 
az-\ri3        az,-\-f3 

is  impossible.     This  is  ensured  by  the  assumption  ad — 13}' ^0. 

We  determine  now  how  many  elements  are  unchanged  by  the 
substitution  u.     An  index  z  can  only  remain  unchanged  by  u  if 

E)  rz'-{-{S—a)z—l3=0     (mod.p). 

There  are  accordingly  four  distinct  cases: 

a)       The  two  roots  of  E)  are  imaginary.     This  happens  if 


(^') 


Tl  {ad—^r=±l) 


is  a  quadratic  non-remainder  (mod.  p).     The  corresponding  substi- 
tutions affect  all  the  elements  Xq,Xi,X2j  .  .  .  x^_i,  x^. 

b)  The  two  roots  of  E)  coincide.     This  happens  if 

(^)t1=0     (mod.p)     {ad-l3r=±l). 

The  corresponding  substitutions  leave  one  element  unchanged. 

c)  The  two  roots  of  E)  are  real  and  distinct.     This  happens  if 

is  a  quadratic  remainder  (mod.  p).     The  corresponding  substitutions 
leave  two  elements  unchanged. 

d)  The  equation  E)  may  vanish  identically.     This  happens  if 

r=0,  13=0,  a  =  d     (mod.  p). 


CERTAIN    SPECIAL    CLASSES    OF    GROUPS.  155 

The  corresponding  substitution  leaves  all  the  elements  unchanged. 
Finally  we  observe  also  that 


M) 


.«^  +  ^ 


rz  +  d 


z 


z 


{ra^-^^r,)z  +  {r^,  +  ^^,) 


n^  +  ^i 

N)  (ad—i^r)  («i^i — /5i  n)  =  (««i  +  M  (r/5i  +  ^^0 

-(«/5i+W(r«i  +  ^ri). 

We  proceed  now  to  collect  our  results.  If  we  take  «  not  ^0 
(mod.  p),  and  /5  and  y  arbitrarily,  then  for  each  of  the  (p  —  l)p^ 
resulting  systems  we  obtain  two  solutions  of  D').  Since  however 
there  are  always  two  systems  of  coefficients  which  give  the  same 
substitutions  u,  We  have  in  all,  in  the  present  case,  p^ — p^  substitu- 
tions. Again,  if  we  take  «^0  (mod.  p)  and  ^5  arbitrarily,  then 
restricting  /?  to  the  values  1,2,  ..  .p  —  1,  we  obtain  from  D')  for 
every  system  a,d,^  two  values  of  y;  but  as  two  systems  of  coef- 
ficients give  the  same  u,  we  have  in  this  case  p{p  —  1)  substitutions. 

There  are  therefore  in  all  p^ — i>  =  (p  +  l)p(p  —  1)  fractional 
linear  substitutions  (mod.  p).     From  M)  it  appears  that  these  form 

a  group.      Among   them   there  are  ^^ ^- — — -  substitutions 

which  correspond  to  the  upper  sign  in  D').  From  M)  and  N)  it  is 
clear  that  these  also  form  a  group.  This  latter  group  is  called  ^^the 
group  of  the  modular  equations  for  p^\* 

Both  groups  contain  only  substitutions  which  affect  either  p  -\-  1, 
or  p,  or  p — 1  elements,  or  no  element.  Those  substitutions  which 
leave  the  element  x^  unchanged,  for  which  accordingly  y^O,  form 
the  metacyclic  group  of  §  134.  As  the  latter  is  two- fold  transitive^ 
it  follows  (Theorem  XIII,  Chapter  IV)  that  the  group  of  order 
{P  +  1)P  {P  —  1)  is  three-fold  transitive. 

Theorem  III.  The  fractional  linear  substitutions  (mod.  p) 
form  a  group  of  degree  p  +  1  and  of  order  {p -\- 1)  p  [p  —  \\ 
Those  of  which  the  determinants  are  quadratic  remainders  {mod.  p) 

form  a  subgroup  of  order  — ^      ^  ^^^  group  of  the  modu- 

lar  equations  for  p.  If  any  substitution  of  these  groups  leaves 
more  than  tivo  elements  unchanged,  it  reduces  to  identity.  The 
first  of  the  two  groups  is  three  fold  transitive. 

*  Cf.  J.  Gierster:  Math.  Ann.  XVIII,  p.  319. 


156  THEORY    OF    SUBSTITUTIONS. 

To  construct  a  function  belonging  to  the  group  of  the  fractional 
linear  substitutions,  we  form  first  as  in  §  133,  a  function  V']  of  the 
elements  Xq,  x^,  X2,  .  .  .  Xp_-^  which  belongs  to  the  group  of  substi- 
tutions 

t=\z  ,3z-^a\     (mod.p).     (a  =  0,  1,  2,  ...p— 1;  /?=:1,2,  ...p  — 1) 

The  substitutions  u,  applied  to  ^'1 ,  produce  p-\-l  values 

^11  ^2)  •  •  •  ^p  +  n 
which  these  substitutions  merely  permute  among  themselves.    Ac 
cordingly,  if  ^'  is  any  undetermined  quantity,  the  function 

S  =  (¥-  f\)  {¥-  ¥,)...  {¥-  !f;^,) 
belongs  to  the  given  group. 

§  137.  We  have  now  finally  to  turn  our  attention  to  those 
groups  all  the  substitutions  of  which  are  commutative. 

We  employ  here  a  general  method  of  treatment  of  very  exten- 
sive application.* 

Suppose  that  0\  d'\  0'" .  .  .  are  a  series  of  elements  of  finite 
number,  and  of  such  a  nature  that  from  any  two  of  them  a  third 
one  can  be  obtained  by  means  of  a  certain  definite  process.  If  the 
result  of  this  process  is  indicated  by  /,  there  is  to  be,  then,  for 
every  two  elements  0\  0",  which  may  also  coincide,  a  third  element 
d"\  such  that  f{0'  0")  =  d'".  We  will  suppose  further  that 
f{o\e")=f{o",o'), 
f[_o'J{e",e'")-]=flf{e',d"\d'"-\, 

but  that,  if  0"  and  6'"  are  different  from  each  other,  then 

f{l>',O")-]^f(0',0"'). 
These  assumptions  having  been  made,  the  operation  indicated  by  / 
possesses  the  associative  and  commutative  property  of  ordinary 
multiplication,  and  we  may  accordingly  replace  the  symbol  f{0',  0") 
by  the  product  0'  6"^  if  in  the  place  of  complete  equality  we  employ 
the  idea  of  equivalence.     Indicating  the  latter  relation  by  the  usual 

sign  00 ,  the  equivalence 

O'd^'coO'" 

is,  then,  defined  by  the  equation 

f{d\d")  =  e"'. 

*L.  Kronecker:  Monatsber.  d.  Berl.  Akad.,  1870,  p.  881.  The  following  is  taken 
for  the  most  part  verbatim  from  this  article. 


CERTAIN  SPECIAL  CLASSES  OF  GROUPS.  l57 

Since  the  number  of  the  elements  0^  which  we  will  denote  by  n, 
is  assumed  to  be  finite,  these  elements  have  the  following  properties : 

I)  Among  the  various  powers  of  an  element  0  there  are  always 
some  which  are  equivalent  to  unity.  The  exponents  of  all  these 
powers  are  integral  multiples  of  one  among  them,  to  which  0  may 
be  said  to  belong. 

II)  If  any  0  belongs  to  an  exponent  v,  then  there  are  elements 
belonging  to  every  divisor  of  v . 

III)  If  the  exponents  p  and  ^,  to  which  0'  and  0"  respectively 
belong,  are  prime  to  each  other,  then  the  product  0'  0"  belongs  to  the 
exponent  fxr. 

IV)  If  nj  is  the  least  common  multiple  of  all  the  exponents  to 
which  the  n  elements  0  belong,  then  there  are  also  elements  which 
belong  to  n^ . 

The  exponent  n,  is  the  greatest  of  all  the  exponents  to  which 
the  various  elements  belong.  Since,  furthermore,  ii^  is  a  multiple 
of  every  one  of  these  exponents,  we  have  for  every  0  the  equival- 
ence   (>"'  CO  1 

§  138.  Given  any  element  0^  belonging  to  the  exponent  n^,  we 
may  extend  the  idea  of  equivalence,  and  regard  any  two  elements 
0'  and   0"  as  ^^ relatively  eqiiivalenV   when  for  any  integral  value 

of  k 

O'.O^^coO" 

We  retain  the  sign  of  equivalence  to  indicate  the  original  more  lim- 
ited relation. 

If  now  we  select  from  the  elements  0  any  complete  system  of 
elements  which  are  not  relatively  equivalent  to  one  another,  this 
subordinate  system  satisfies  all  the  conditions  imposed  on  the  entire 
system  and  therefore  possesses  all  the  properties  enumerated  above. 
In  particular  there  will  be  a  number  ^2 ,  corresponding  to  n^ ,  such 
that  the  ^2*^  power  of  every  0  of  the  new  system  is  relatively  equiv- 
alent to  unity,  1  e.,  O^'-oo  (>■!'.  Again  there  are  elements  0,,  in  the 
new  system  of  which  no  power  lower  than  the  n.}^  is  relatively 
equivalent  to  unity.  Since  the  equivalence  ^"^  oo  1  holds  for  every 
element,  and  consequently  a  fortiori  eyerj  0"-  is  relatively  equiva- 


158  THEORY    OF    SUBSTITUTIONS. 

lent  to  unity,  it  follows  from  I)  than  n^  is  equal  to  n^  or  is  a  multiple 

of   Wg. 

If  now 

and  if  both  sides  are  raised  to  the  power  — ,  we  obtain,  writing 

k 
—  =  m,  the  equivalence 

From  this  it  follows  that,  since  6^  belongs  to  the  exponent  nj ,  m  is 
an  integer  and  k  is  therefore  a  multiple  of  ng.  There  is  therefore 
an  element  <?2)  defined  by  the  equivalence 

e^e^'^ooo,,    or    d^ooo„d^''i-'^ 

of  which  the  n^^  power  is  not  only  relatively  equivalent,  but  also 
absolutely  equivalent  to  unity.  This  element  belongs  both  rela- 
tively and  absolutely,  to  the  exponent  ng,  for  we  have  the  relation 

Proceeding  further,  if  we  now  regard  any  two  elements  6'  and 
d"  as  relatively  equivalent  when 

d'  d^^'d^^cod", 
we  obtain,  corresponding  to  0^ ,  an  element  6^  belonging  to  the  expo- 
nent ^3,  where  n^  is  equal  to  n^  or  a  divisor  of  n^',  and  so  on.     We 
obtain  therefore  in  this  way  a  fundamental  system  of    elements 
^1 )  ^2 )  ^3  >  •  •  •  which  has  the  property  that  the  expressions 
d,\d^o.O^^ . .  ,     {h,=  1,  2,  ..  .n) 

include  in  the  sense  of  equivalence  every  element  0  once  and  only 
once.  The  number  ni ,  rig ,  ng ,  .  .  . ,  to  which  the  elements  ^i ,  ^2  >  <^3  ?  •  • . 
belong,  are  such  that  every  one  of  them  is  equal  to  or  is  a  multiple 
of  the  next  following.  The  product  Wj  tig  W3 .  .  .  is  equal  to  the 
entire  number  n  of  the  elements  ^,  and  this  number  n  accordingly 
contains  no  other  prime  factors  than  those  which  occur  in  the  first 
number  Wj . 

§  139.  In  the  present  case  the  elements  0  are  to  be  replaced  by 
substitutions  every  two  of  which  are  commutative.  The  number  n 
of  the  elements  0  becomes  the  order  r  of  the  group.  We  have 
then 


CERTAIN    SPECIAL    CLASSES    OF    GROUPS.  159 

Theorem  IV.  If  all  the  substitutions  of  a  group  are  com- 
mutative^ there  is  a  fundamental  system  of  substitutions  s^^s^^s^,,  . . 
which  possesses  the  property  that  the  products 

siH*%*3...    (hi  =  1,2,, .  .n) 

include  every  substitution  of  the  group  once  and  only  once.  The 
numbers  r^^r^^r^,  .  .  .  are  the  orders  of  Si,S2,s^,  .  .  .  and  are  such 
that  every  one  is  equal  to  or  is  divisible  by  the  next  following.  The 
product  of  these  orders  r^,r<i,r^,  .  .  ,  is  equal  to  the  order  r  of  the 
igroup. 

The  number  r^  is  determined  as  the  maximum  of  the  orders  of  the 
several  substitutions.  On  the  other  hand  the  corresponding  substi- 
tution §1  is  not  fully  determined,  but  may  be  replaced  by  any  other 
substitution  s/  of  order  r^ .  According  then  as  we  start  from  s^  or 
Si',  the  values  of  rg ,  rg ,  .  .  .  might  be  difPerent.  We  shall  now 
show  that  this  is  not  the  case. 

In  the  first  place  it  is  plain  that  if  several  successive  s's  belong 
to  the  same  exponent  r,  these  s's  may  be  permuted  among  them- 
selves, without  any  change  in  the  r's.  Moreover,  every  s^  can  be 
replaced  by  Sa^Sa^]' Sa-^2^ .  .  .  without  any  change  in  the  r's,  pro- 
vided only  that  //.  is  prime  to  r^. 

If  now  the  given  group  can  be  expressed  in  the  two  difPerent 
forms 

s.hs.'^^s,^-^  ...{K  =  l,2,...  r,),       (T,^ff-M,^  ...(/?,,=  1,  2,  ..  .  />,), 
then  p^  =  r^,  and 

Since  (t^  belongs  to  7\ ,  at  least  one  of  the  exponents  p.,  v,  .  .  .  must 
be  prime  to  r^ .  From  the  first  remark  above,  we  may  assume  that 
this  is  /x,  and  from  th6  second  it  follows  that  the  group  can  also  be 
expressed  by 

^i^s/V=^  ...      (/I,  =  1,  2,  .  .  .  r,;  p,  =  r,). 

Consequently  the  groups 

s^s/3  ...(h,  =  l,2,...  r,),        <r,^^<T,^^ .  .  .  (/I,  =  1,  2,  .  .  .  />,) 
are  identical.      From  this  it  follows,  as  before,  that  p^  —  r^,  and 
80  on. 

Theorem  V.     The  numbers   r^^r^^r^,  .  .  .   are  invariant  for 

a  given  group.* 

*This  theorem  is  due  to  Frobenius   and  Stickelberger.   cf.  their  article:  Uber 
Gruppen  mit  vertauschbaren  Elementen;  Crelle,  86,  pp.  217-262. 


CHAPTER  VIII. 


ANALYTICAL  KEPRESENTATION  OF  SUBSTITUTIONS. 
THE  LINEAR  GROUP. 

§  140.  In  the  preceding  Chapter  we  have  met  with  a  fourth 
method  of  indicating  substitutions,  which  consisted  in  assigning  the 
analytic  formula  by  which  the  final  value  of  the  index  of  every  x  is 
determined  from  its  initial  value.  Thus,  if  the  index  z  of  every  x^ 
is  converted  by  a  given  substitution  into  (p{z)^  so  that  x^  becomes 
Xa,(^),  the  substitution  is  completely  defined  by  the  symbol 

s^\z  (p{z)\. 

Obviously  not  every  function  can  be  taken  for  (f{z)^  tor  it  is  an 
essential  condition  that  the  system  of  indices  <p(l),  v(2),  .  .  .  <p{n) 
shall  all  be  different  and  shall  be  identical,  apart  from  their  order, 
with  the  system  1,  2,  3,  .  .  .  n.  On  the  other  hand  it  is  readily 
shown  that  every  substitution  can  be  expressed  in  this  notation. 
For  if  it  is  required  that 

cr(l)=*i,  (p{2)  =  i„  .  .  .(p{n)  =  K, 

we  can  construct,  by  the  aid  of  Lagrange's  interpolation  formula, 
from 

F{1)  =  (z—l)  {z—2) .  .  .  (z—n) 

a  function  <s{z)  which  satisfies  the  conditions,  viz: 

'^K^)  -  '^ j^^'(z){z—iy    - F'{z){z—2)  ^  '  '  '^  "F'{z){z  —  7iy 

This  function  is  of  degree  n  —  1  in  z.  It  is  evident  that  there  are 
an  infinite  number  of  other  functions  <p{z)  which  also  satisfy  the 
required  conditions. 

§  141.  If  n  is  a  prime  number  p,  we  can  on  the  one  hand, 
diminish  the  restrictions  imposed  on  <p  by  permitting  the  indices 
1,  2,  8,  .  .  .p  to  be  replaced  by  any  complete  system  of  remainders 
(mod.  p),  so  that  indices  greater  than  p  are  also  allowable.     And  on. 


ANALYTICAL  REPRESENTATION  OF  SUBSTITUTIONS.         161 

the  other  hand  we  can  depress  every  form  of  (p(z)  to  the  degree 
n  —  1,  since  z^^^z  (mod.  p)  for  all  values  of  z. 

In  particular,  we  have  in  this  case  F{z)^z^'  —  z,  and 
F'{z)=pz^''  —  lEE^  —  l. 

For  n  =  p,  the  functions  (p{z)  which  are  adapted  for  the  expres- 
sion of  a  substitution,  for  which  therefore  ^(0),  (f(i),  .  .  .  (p(p  —  1) 
form  a  complete  system  of  remainders  (mod.  p),  are  defined  by  the 
following  theorem : 

Theorem  I.  In  order  that  \z  <pz\  may  express  a  substitu- 
tion of  p  elements,  it  is  necessary  and  sufftcient  that  after  (p{z)  and 
its  first  p — "2  powers  have  been  depressed  to  the  degree  p  —  1  by 
means  of  the  congruence  z^'^^i^z  {mod.  p),  and  after  all  multiples  of 
p  have  been  removed,  these  2^  —  1  potvers  of  <p{z)  should  all  reduce  to 
the  degree  p — 2.  * 

Let 

0{z)  =  Ao-^A,z-^A,z'+  .  .  .  -^A^,_,z^-' 

be  any  integral  function  (mod.  p),  and  suppose  that 

[^(^]'"  =  ^,/'"-  +  ^\'"'^z  +  A./'"-V  +...-[-  A^,_^'"'^z''  -'     (mod.  p). 
Since  for  every  a  <  p  —  1 

0»  +  l'^  +  2«+  .  .  .  +(p  — 1)«ee:0     (mod.  21), 
we  have 

S)  WO)]-  +  [c.(l)]'"+  .  .  .  +[c.(p-l]'»EE^(p-l)^^,_,('») 

^— A,, __/'")     (mod.  p). 

If  now  <f(z)  is  adapted  for  the  expression  of  a  substitution,  then,  as 
<p{0),<p{l),...(p(2)  —  1)  form  a  complete  system  of  remainders 
(mod.  p),  we  conclude  that  for  m  <  p  ^  1 

-A„_i''")e^O     (mod.  p). 
This  is  therefore  a  necessary  condition. 

Conversely,  if  this  condition  is  satisfied  for  a  given  function 
(p{z),  then  since  S)  holds  for  m  =1,2,  ...  p  — 2,  it  follows  from 
formula  B)  of  §  8  that 

[0  —  ^(0)]  [0—^(1)]  .  .  .  [z  —  <p{p  —  l)]EEEZ^—aZ—^ 

^(1  —  'J-)z — fi     (mod.  p). 

*Hermite:  Comptes  rend  lis  de  I'Acad^mie  des  Sciences,  57. 
11 


162  THEOEY    OF    SUBSTITUTIONS. 

Accordingly,  if  a==0,  the  linear  congruence  (1  —  a)z  —  /^^(mod.  p) 
is  satisfied  by  thep  integers  ^(0),  ^(1),  .  .  .  <p{p — 1).  These  must 
therefore  all  be  equal,  and,  as  their  sum  is  0,  every  one  of  them  is 
equal  to  0.     But  in  this  case  the  congruence  of  degree  p  —  2 

<p{z)=A^-^A,z-{-A^z^-\-  .  .  .  +A^_,z^-'=0     (mod.  p) 

would  have  p  —  1  different  roots  z  =  0,l,2,  . .  .p  —  l,  which  is 
impossible.     Consquently  a  =  l  and  then  /?  =  0,  that  is 

[^-^0)]  [^-Hi)]  • . .  [z-<p{p-i)y~z--z 

^z{z-l){z-2) . .  .[z-{p-l)], 
and  the  values  ^(0),  ^(1),  .  .  .  <p(p  —  1)  coincide  with  0, 1,  .  .  .p  —  1, 
apart  from  their  order.  It  appears  therefore  that  the  condition 
stated  in  Theorem  I  is  both  necessary  and  sufiicient  to  insure  that 
\z  <pz\  defines  a  substitution. 

§  142.  To  distinguish  the  individual  elements  of  any  system 
we  may  also  employ  several  indices  in  each  case  instead  of  a  single 
index  as  heretofore.  For  example,  in  the  case  of  p^  elements  the 
indices  z  and  u  of  a?^,„  might  each  assume  any  value  from  0  to 
p — 1.     Any  substitution  among  these  p^  elements  could  then  be 

denoted  by 

s=\z,u     <p{z,u),(/'(z,u)\ 

where  <p{Zy  u)  and  (/'{z,  u)  must  satisfy  conditions  similar  to  those  of 
§  140. 

If  n  =  p*  the  elements  could  be  denoted  by 

a7,^,a?,,,...a?,^     (^,  =  0, 1,  2,  . .  .p  — 1), 
and  any  substitution  s  by  the  symbol 

S=\Z,,Z2,....Z,,      <p,{Zi,Z2,...Zk),  (P2{Z,,  Z^y. . .  Zj^),  .  .  .  <Pk{z„Z2,.  . .  Zk)\, 

which  is  to  be  understood  as  indicating  that  every  index  z,-,  is  to  be 
■converted  into  <pi{Zi,Z2i  .  .  .  Zj^.  The  functions  <Pi<,(P2,  •  -  -  *Pk  must 
then  be  so  taken  that  thep*'  different  systems  of  indices  Zi,Z2,  .  .  .z^ 
give  rise  to  p^  different  systems  ^i ,  ^2?  •  •  •  ^fc- 

These  considerations  could  be  further  extended  to  include  the  case 
where  n  contains  several  distinct  prime  factors,  but  as  the  theory 
then  becomes  much  more  complicated,  we  do  not  enter  upon  it  here. 

§  143.  The  simplest  analytical  expressions  for  substitutions  of 
n  =  m*  elements  are  those  of  the  linear  form 


ANALYTICAL    KEPRESENTATION    OF    SUBSTITUTIONS.  163 

1)  Sa,,ao,...a^.~-^\ZuZ.,,  .  .  .Zj,       Z^  A- a^,  Zr^ -{- a^,  .  .  .  Zk+ ^kl  • 

The  a's  are  arbitrary  integers  (mod.  m).  They  can  therefore  be 
selected  in  m*'  different  ways,  so  that  there  are  m^  substitutions  of 
this  type.     Again,  since 

2)  Sai,  02,  ...  tti,  •  %,i3.2,  ■••/3;fc  ==^V  +  ^l,a2  +  ^2.  •••  aA  +  Pfc» 

these  arithmetic  substitutions  *  form  a  group  of  order  and  degree 
m^  This  group  is  transitive,  since  the  «'s  can  be  so  chosen  that 
any  given  element  a^^^ ,  ^3 , . . .  s^  ^^  replaced  by  any  other  element 
x^^ ,  C2,.--iic'     -^^^  *^^^  purpose  we  need  only  take 

«i  =  ,1  0],  «2  ^^  '2 'S'2  5    •   •   •   '^i-  =  '/.•  ^k- 

There  is  only  one  substitution  of  the  group  which  produces  this 
result. 

In  order  that  an  arithmetic  substitution  may  leave  any  element 
X  unchanged,  it  is  necessary  that 

</j  =  0,     a^  =  0,  .  .  .  a;fc  =  0     (mod.  m) 

Such  a  substitution  leaves  all  the  elements  unchanged,  and  there- 
fore reduces  to  identity.  The  present  group  is  therefore  included 
in  the  groups  ^2  considered  in  the  preceding  Chapter. 

By  the  continued  application  of  the  formula  2)  we  obtain 

^ttj  ,  tto ,  .  .  .  a;^.  ^  ^1  ,  0 ,  .  .  .  0^  •  ^0  ,  1  .  .  .  .  0   "...  ^0 ,  0 ,  ...  1    * 

so  that  we  may  define  the  group  by 

^)  ^  ^^  ]  ^1 ,  0 ,  . . .  0  ,    ^0  ,  1 ,  . . .  0  J  •  •  •  ^0 , 0  ,  . . .  M  • 

The  substitutions  contained  in  the  parenthesis  are  all  commuta- 
tive, and  the  same  property  consequently  holds  for  all  the  substitu- 
tions of  G'. 

§  144.     We  determine  now  the  most  general  form  of  the  substi- 
tutions 

t—\Zi,Z2,...Zk      <Pi{Zi,Z2,. . .  Zf,),  <P2{Z,,  Z^, .  . .  Zk),. . .  M^u  z^,. . .  Zk)\, 

which  are  commutative  with  G,  for  which  therefore 

It  is  obviously  sufficient  to   take    for    Saj , . . .    the  several  genera- 
ting substitutions    given    in    3).      The    substitution    ^~^Si,o,...o^ 
Cauchy:  Exercices  III.,  p.  232. 


164  THEORY    OF    SUBSTITUTIONS. 

replaces  ^Az^.z^^,  .  .  .  %)  by  <Pk{z^  +  1,2.,  .  .  .  2;^.).  Consequently, 
taking  /  =  1, 2, 3,  . .  .  /c  we  have 

<pAzj  +  1,2:.,,...  Zf^)  =  <f},{zi ,z.^,  .  .  .  Zj,)  -[-  «A     (mod.  m). 

Similarly  in  the  case  of  the  substitutions  So^  1,  ...o?  •  •  •  So,  0.  ...1  we 
obtain 

<P\{Zi ,Z2-\-h  .  .  .  z^)^ <px{z^ ,z.2,  .  .  .Zt)-\- 6a     (mod.  m) 

(fx{z^ ,  2:, ,  .  .  .  2:a.  +  1)  =  ^a(2i  ,  22 ,  ...  2a.)  +  Ca     (mod.  m). 

From  these  congruences  it  appears  at  once  that  the  cta's  are 
linear  functions  of  the  2's,  having  for  their  constant  terms 
f^A  =  v^a(0,  0,0,...  0).  The  remaining  coefficients  are  then  readily 
found.     In  fact,  we  have 

crA(2i,22,  .  .  .2A.)  =  aA2i+?7A22+  •  •  •  +Ca2;,.  +  0a, 

and  therefore 

^  =  I  2, ,  22 ,  ,  .  .  2i.     ai2i  +  b^z.,  +  .  .  .  +  Ci2a.  +  '^1 , 

a,,2j  +  h.z.  +  .  .  .  +  c.^Zt,  +  ^2, .  .  .  |. 
Conversely  all  substitutions  of  this  type  transform  the  group  G  into 
itself.     Thus,  for  example,  t  transforms  -Si ,  0 , 0 , . . .  u  into 

|ai2a  +  6i22+  .  .  .  +C,2,.+  0\,  .  .  . 

a,{z,  +  1)  +  &i2'2  +  .  .  .  +  c,2,.  +  ^, , .  .  .  I, 

* 

i.  e.,  into  the  arithmetic  substitution 

By  left  hand  multiplication  by 

SSi ,  5,,  . . .  6/.  ^  ^  =  S      5i  ,      So  ,  . . .  -  5a,- 

we  can  reduce  t  to  the  form 

t'  =  \z,,Zo,  ...  z,.     a^z,  +  6i22  +  .  .  .  4-  c,2a.,  a22,  +  b,z,  +  .  .  .  +  c^z,., 

.  .  .  aA,2i  +  6,.22  +  .  .  .  +c^Zk\. 
Such  a  substitution  is  called  a  geometric  suhstitution.  * 

We  proceed  to  examine  this  type.     We  have  already  demon- 
strated 

Theorem  II.     All  geometric  iiiibstitutions  and  their  combi- 
nations with  the  arithmetic  substitutions,  g,nd  no  others,  are  com- 
mutative with  the  group  of  the  arithmetic  substitutions. 
•Cauchy:  loc.  cit. 


ANALYTICAL    EEPRESENTATION    OF    SUBSTITUTIONS.  165 

§  145.  We  have  first  of  all  to  determine  whether  the  con- 
stants a\,b\,  .  .  .  Ca.  can  be  taken  arbitrarily.  They  must  certainly 
be  subjected  to  one  condition,  since  two  elements  a?^^ ,  ^„ , . . .  ^^^  and 
a?^j ,  ^o , . . .  c>^  must  not  be  converted  into  the  same  element  unless  the 
indices  Zi,Z2,  .  .  .  z^  coincide  in  order  with  Ci,  C25  •  •  •  '*•  More 
generally,  given  any  system  of  indices  Ci,  C2,  .  .  .  C^.,  it  is  necessary 
that  from 

a^Zi  4-  b^z,  -j- . . .  -h  Ci% EEE  Ci ,  a.2Zi  +  622^2  +  •  •  •  +  ^2% = Cg ,  •  •  •  (mod.  m) 
the  indices  Zi,  z.,,  .  .  .  z^  shall  be  determined  without  ambiguity. 
In  other  words,  the  m''  systems  of  values  z  must  give  rise  to  an 
equal  number  of  systems  of  values  C .  The  necessary  and  sufficient 
conditions  for  this  is  that  the  congruences 

ayb^  +  612:2  + . . .  +  c^Zj,. ^ 0,  a^z^  -\-  b^z^  + . .  •  +  C2Z;,-^0 , . . .      (mod.  m) 

shall  admit  only  the  one  solution  ^i  =  0,  Z2  =  0,  .  .  .  Zj,  =  0.  If  the 
determinant  of  the  coefficients  is  denoted  by  J,  these  congruences 
are  equivalent  to 

J  .  2:1^0,  J  .  2:2:^0,  . .  .  J-Z),^0     (mod.  m). 

The  required  condition  is  therefore  satisfied  if  and  only  if  J  is 
prime  to  m.     We  have  then 

Theorem  III.     In  order  that  the  symbol 

t=\z,,Z2,...z^.,    aiZi-i-b,Z2-Jr...+CiZk,  a22:i  +  622^2  +  ..  • +^22:^,. ..  I 

(mod  m) 
may  denote  a  (geometric)  substitution,  it  is  necessary  and  sufficient 

that 

tti,  61,  .  .  .Ci 

J   __        tt2  5    ^2  )   •  •   •  ^2 
Otfc)   O^,   .   .   .  C;!; 

should  be  prime  to  the  modulus  m. 

§  146.  From  this  consideration  it  is  now  possible  to  determine 
the  number  r  of  the  geometrical  substitutions  cQ^esponding  to  a 
given  modulus  m. 

We  denote  the  number  of  distinct  systems  of  p  integers  which 
are  less  than  m  and  prime  to  m  by  [m,  /?].  It  is  to  be  understood 
that  any  number  of  the  />  integers  of  a  system  may  coincide. 


166  THEORY    OF    SUBSTITOTIONS. 

Suppose  N  to  be  the  number  of  those  geometric  substitutions 
1,  ^2 )  ^3  J  •  •  •  which  leave  the  first  index  Zi  unchanged.  If  then  r^  is 
any  substitution  which  replaces  z^  by  ai^i  + 612:2+  •  •  •  H~Ci%,  then 
r2,  t^r^,  t^r^,  . .  .  are  all  the  substitutions  which  produce  this  effect, 
and  these  are  all  different  from  one  another.  Similarly,  if  r^  replaces 
Zi  by  a/ Zi-{-b/z2 -{-...  -\- c/z^,  then  '3,  #2^35  te?  •  •  •  are  all  the 
substitutions  which  produce  this  effect,  and  these  are  all  different, 
and  so  on.  We  obtain  therefore  the  number  r  of  all  the  possible 
geometric  substitutions  by  multiplying  N  by  the  number  of  substi- 
tutions   1,  Tg,  Tg,  .  .  . 

'  The  choice  of   the  systems    ai,  61,  .  .  .  Cj;  ai,bi,  .  .  .  c/;  .  .  .    is 

limited  by  the  condition  that  that  the  integers  of  a  system  cannot 

have  a  same  common  factor  with  m.     There  are  therefore  [m,  k] 

such  systems,   and  an   equal  number  of  substitutions    1,  T2,r3,... 

Consequently 

r=  [m,  k^N. 

The  substitutions  t  are  of  the  form 

\zi,Z2,.  .,Zk   Zi,a2Zi-^b2Z2-\- C.2ZJ,,  ...akZi  +  b^2+'--  +  c^k\ 

(mod.  m). 
Since  a2^a^,  .  .  .  a^  do  not  occur  in  the  expression  of  the  discrimi- 
nant /J,  these  integers  can  be  chosen  arbitrarily,  ^.  e.,  in  m^  "^  dif- 
ferent ways.     The  6a  ,  •  •  •  Ca  are  subject  to  the  condition  that 

62 ,  .  .  .  C3 

6fc,. . .Cfc 

must  be  prime  to  m.     If  the  number  of  systems  here  admissible  is 
r',  we  have 

The  number  r  has  the  same  significance  for  a  substitution  of 
k — 1  indices  (mod.  m)  as  r  for  k  indices.     Consequently 

r  =  [m,  k~\7n!'~^ [m,  k — 1]  m*"" V", 
and  so  on.     We  obtain  therefore  finally 

r  =  [m,  k']m!'-'  [m.  A;  — l]m^  -^  .  .  [m,  2]r(^-^), 

where   r<^^^>  corresponds  to  a   single  index,  and  therefore  r(*~') 
=  [m,  1] .      Hence 


ANALYTICAL    REPKESENTATION    OF    SUBSTITUTIONS.  167 

4)  r  =  [m,  A;]  m^~^  [m,  k —  1]  m''~'^ .  .  .  [m,  2] m [m,  1]  . 

The  evaluation  of  [m,  A;]  presents  little  difficulty.  We  limit  our- 
selves to  the  simple  case  where  m  is  a  prime  number  p,  this  being 
the  only  case  which  we  shall  hereafter  have  occasion  to  employ.  We 
have  then  evidently 

5)  [p,p]=P'—l, 

since  only  the  combination  0,  0 ,  ...  0  is  to  be  excluded.     By  the 
aid  of  5),  we  obtain  from  4) 

6)  r  =  (p*-l)p'=-(p''-'-l)p'-^ . . .  {p^-l)p(p--l) 

=  ip"-^)  iP'-p)  iP'-p')  .  ■  ■  (p'-p"-').* 
§  147.     The    entire    system    of     the    geometric    substitutions 

(mod.  m)  forms  a  group  the  order  of  which  is  determined  from  4) 

or  from  6).     This  group  is  known  as  the   linear  group  (mod.  m). 

If  the  degree  is  to  be  particularly  noticed,  we  speak  of  the  linear 

group  of  degree  m/". 

It   is  however  evident  that  all  the  substitutions  of  this  group 

leave  the  element  Xq,q,  .  .  .  o  unchanged.     For  the  congruences 

ci'i^i  +  b,Z2  +  .  .  .  -h  Ci%=2;i ,     a^z.i  +  hz^  +  h^2+  •  •  •  +  C2%  =  22,  •  •  • 

(mod.  m) 

have  for  every  possible  system  of  coefficients  the  solution 

^lE^O,     z^^O,  .  .  .  Zk^^O     (mod.  m). 

We  shall  have  occasion  to  employ  the  linear  group  in  connection 
with  the  algebraic  solution  of  equations. 

Theorem  IV.  The  group  of  the  geometric  substitutions 
(mod.  m),  or  the  linear  group  of  degree  wJ"  is  of  the  order  given  in 
4).  Its  substitutions  all  leave  the  element  a?o,o)  ...o  unchanged.  It 
is  commutative  with  the  group  of  the  arithmetic  substitutions. 

♦Galois:  Liouville  Journal  (1)  XI,  1846,  p.  410. 


PART   II. 

APPLICATION  OF  THE  THEOKY  OF  SUBSTITUTIONS  TO 
THE  ALGEBRAIC  EQUATIONS. 


CHAPTER  IX. 

THE    EQUATIONS    OP    THE    SECOND,    THIRD    AND    FOURTH 
DEGREES.    GROUP  OF   AN   EQUATION.     RESOLVENTS. 

§  148.     The  problem  of  the  algebraic  solution  of  the  equation 
of  the  second  degree 

1)  ^  x'^ — c^x-\-C2  =  0 

can  be  stated  in  the  following  terms:  From  the  elementary  sym- 
metric functions  c^  and  c.2  of  the  roots  Xi  and  x.,  of  1)  it  is  required 
to  determine  the  two- valued  function  Xi  by  the  extraction  of  roots.* 
Now  it  is  already  known  to  us  (Chapter  I,  §  13)  that  there  is  always 
a  two- valued  function,  the  square  of  which,  viz.,  the  discriminant, 
is  single-valued.     In  the  present  case  we  have 

J  =  (xi  —  x^y  =  (a?i  +  ^if — 4:XiXo  =  c^ — 4c2 , 
sjA  —  (x^  —  X2)  —  \/ c^ — 4c2. 
Since  there  is  only  one  family  of  two -valued  functions,  every  such 
function  can  be  rationally  expressed  in  terms  of   sj  ^.     For  the 
linear  two-valued  functions  we  have 

and  in  particular,  for  «!  =  1,  «2  =  0,  and  for  «!  =  0,  «2  =  1 


Ci        ,       -  /        Tt  \  Ci 


a?i==Y  +  f Vci'— 4c2,    a?2=^— jVci'— 4c2. 

*C.  G.J.  Jacobi:  Observatiuueulae  ad  theoriain  aequationiim  pertiuentes.  Werke, 
Vol.  Ill;  p.  269.  Also  J.  L.  Lagrange:  R6flexions sur  la  resolution  alg6brique  des  equa- 
tions.   Oeuvres.  t.  III,p.  205. 


ELEMENTARY    CASES GROUP    OF    AN    EQUATION RESOLVENTS.  169 

§  149.     In  the  case  of  the  equation  of  the  third  degree 


/o*^  v^3 


the  solution  requires  not  merely  the  determination  of  the  three- 
valued  function  x^ ,  but  that  of  the  three  three -valued  functions 
Xi,X2,  x.^.     With  these  the  3! -valued  function 

is  also  known,  and  conversely  x^^x^^x.^  can  be  rationally  expressed 
in  terms  of  c.  We  have  therefore  to  find  a  means  of  passing  from 
the  one-valued  functions  Cj ,  C2 ,  C3  to  a  six- valued  function  by  the 
extraction  of  roots. 

In  the  first  place  the  square  root  of  the  discriminant 

J  =:  (a?j — x^\x^ — a?3)^(a?2  —  x^^  —  — 27c3^  +  I8C3C2C1 — 4c3C,^ 
furnishes  the  two-valued  function 

in  terms  of  which  all  the  two-valued  functions  are  rationally  expres- 
sible. The  question  then  becomes  whether  there  is  any  multiple- 
valued  function  of  which  a  power  is  two -valued.  This  question  has 
already  been  answered  in  Chapter  III,  §  59.  The  six-valued  func- 
tion 

9x-^x-V ^^2  +  ^'^3    ^>  =  y 2 J, 

on  being  raised  to  the  third  power,  gives 


again,  if  ^2  is  obtained  from  ^^  by  changing  the  sign  of  \/  —  3 » 
we  have 

Accordingly 

Xi-\-w%-{-ajx^=  4/i(;Si  +  3/\/— 3^, 
Xi-{-wx2-{-(o^x^—  \/i(Si — 3V — 3J. 
Combining  with  these  the  equation 

£Cl  -j-  £^2  -j-  a?3  =  Cj  , 


170  THEORY    OF    SUBSTITUTIONS. 

and  observing  that 

l  +  «>  +  .^'  =  0, 

we  obtain  the  following  results 


^3z=i[cj  +  «>2\^i(;Si  +  3V^^^^3^  +  ">^i('5i  — 3V^^^^3J)]. 
The  solution  of  the  equation  of  the  third  degree  is  then  complete. 
§  150.     In  the  case  of  the  equation  of  the  fourth  degree 

it  is  again  only  the  one-valued  functions  Cj ,  C2 ,  Cg ,  c^  that  are  known. 
From  these  we  have  to  obtain  the  four  four- valued  functions 
a?i ,  072,  a?3,  iz^4,   and  with  them  the  24- valued  function 

by  the  repeated  extraction  of  roots. 

In  the  first  place  the  square  root  of  a  rational  integral  function 
of  Ci,C2,C3,C4  furnishes  the  two- valued  function  y'j.  Again,  we 
have  met  in  §  59  with  a  six-valued  function 

(p  =  {x^x^  +  x^x^  +  w(a?ia?3  +  x<pc^  +  oy\x-^x^  +  x.^x^ , 

the  third  power  of  which  is  two-valued  and  therefore  belongs  to  the 
family  of  y' J.  We  can  therefore  obtain  <p  by  the  extraction  of  the 
cube  root  of  a  two-valued  function.  ^  being  determined,  every 
function  belonging  to  the  same  family  is  also  known.  The  group 
of  <p  is 

G  =  [1,  (x^X2)  (x^x^),  (x^x^)  (x^x^),  (x^x^)  (a?2a?3)] ;     />  =  6,  r  =  4. 
To  this  same  group  belongs  the  function 

which  can  therefore  be  rationally  obtained  from  <p,  while  0,  which 

can  be  obtained  from  ^^  by  extraction  of  a  square  root,  belongs  to 

the  group 

H=\l,{x,x^{x^x^'\\     />  =  12,  r  =  2. 

0  may  therefore  be  regarded  as  known.     Finally 

y2  —  [a^{x^  —  x^)  4-  a2(a?3  —  X.yf,      r  —  ^^{x^  +  X^  +  ^j^X^  +  X^ 


ELEMENTARY    CASES GROUP    OF    AN    EQUATION ^^"rESOL VENTS.  171 

can  be  rationally  expressed  in  terms  of  4'',  and  ;>:  is  a  24-valued 
function.  All  rational  functions  of  the  roots,  and  in  particular  the 
roots  themselves  can  then  be  rationally  expressed  in  terms  of  /. 
To  determine  the  roots  we  may,  for  example,  combine  the  four 
equations  for  /  and  r  in  which  a^  z=z  ±  a^  and  fti—  ±  1^2  • 

§  151.  In  attempting  the  algebraic  solution  of  the  general 
equations  of  the  fifth  degree  by  the  same  method,  we  should  not  be 
able  to  proceed  further  than  the  construction  of  the  two-valued 
functions.  For  we  have  seen  in  §  58  that  for  more  than  four 
independent  quantities  there  is  no  multiple- valued  function  of 
"which  a  power  is  two-valued.  It  is  still  a  question,  however,  whether 
the  solTltioh"  of  the  equation  fails  merely  through  a  defect  in  the 
method  or  whether  the  impossibility  of  an  algebraic  solution  resides 
in  the  nature  of  the  problem.  It  will  hereafter  be  shown  that  the 
latter  is  the  case.  We  shall  demonstrate  the  full  sufficiency  of  the 
method  by  the  proof  of  the  theorem  that  every  irrational  function  of 
the  coefficients  which  occurs  in  the  algebraic  solution  of  an  equation 
is  a  rational  function  of  the  roots.  All  the  steps  leading  from  the 
given  one- valued  function  to  the  required  wl-valued  functions  can 
therefore  be  taken  within  the  theory  of  the  integral  rational  func- 
tions of  the  roots.  ^ 

§  152.     We  turn  our  attention  next  to  the  accurate  formulation 
of  the  problem  involved  in  the  solution  of  algebraic  equations. 
Suppose  that  all  the  roots  of  an  equation  of  the  n^^  degree 

1)  /(^)  =  o 

are  to  be  determined.  If  one  of  them  x^  is  known,  the  problem  is 
only  partially  solved.  By  the  aid  of  the  partial  solution  we  can 
however  reduce  the  problem  and  regard  it  now  as  requiring  not 
the  determination  of  the  ?i  —  1  remaining  roots  of  the  equation  of 
the  'n}^  degree  f(x)  =  0,  but  that  of  all  the  n  —  1  roots  of  the 
reduced  equation 

We  have  then  still  to  accomplish  the  solution  of  2).  If  one  of  its 
roots  a?2  is  known,  we  can  reduce  the  problem  still  further  to  that  of 
the  determination  of  the  n  —  2  roots  of  the  equation 


172  THEORY    OF    SUBSTITUTIONS. 

Proceeding  in  this  way  we  arrive  finally  at  an  equation  of  the  first 
degree. 

It  appears  therefore  that  solution  of  an  equation  of  the  n^^ 
degree  involves  a  series  of  problems.  All  of  these  problems  are 
however  included  in  a  single  one,  that  of  the  detei^minatiou  of  a 
single  root  of  a  certain  equation  of  degree  n\ 

Thus,  if  the  entirely  independent  roots  a?i ,  .T2 ,  ...  oc„  of  the 
equation  1)  are  known,  then  the  tz!- valued  function  with  arbitrary 
constants  «i ,  «2 ,  .  .  .  «„ 

4)  c  =  «iXi  -|-  a^X^  +  .  .  .  +  a^X^ 

is  also  known.  Conversely  if  ^  is  known,  every  root  of  the  equa- 
tion 1)  is  known,  for  every  rational  integral  function  of  a?i,a'2,...a?„ 

can  be  rationally  expressed  in  terms  of  the  n!- valued  function  c, 

« 
The  function  c  satisfies  an  equation  of  degree  n\ 

5)  F{S)  =  r'—Ar'-'  +  ...  ±  A„,=(f-c,)(|:-c.)...(f-f„,)  =0, 
the  coefficients  of  which  are  rational  integral  functions  of  those  of 
1)  and  of  «!,  ttg.  .  .  .  «„.  This  equation  is,  in  distinction  from  1), 
a  very  special  one.  For  its  roots  are  no  longer  independent,  as  was 
the  case  \^ith  1),  but  every  one  of  them  is  a  rational  function  of 
every  other,  since  all  the  values  Ij ,  ^2  ?  •  •  •  ^n  \  belong  to  the  group  1 
with  respect  to  a^i,  0^2,  .  .  .  x„.  The  solution  of  5),  and  consequently 
that  of  1)  is  therefore  complete,  as  soon  as  a  single  root  of  the 
former  is  known. 

The  equation  F{^)  —  0  is  called  the  resolvent  equation,  and  f 
the  Galois  resolvent  of  1).  We  shall  presently  introduce  the 
name  "resolvent"  in  a  more  extended  sense. 

§  153.  If  the  coefficients  Cj,  C2,  .  .  .  c„  of  the  equation  1)  are 
entirely  independent,  the  Galois  resolvent  cannot  break  up  into 
rational  factors.  Conversely,  if  the  Galois  resolvent  does  not  break 
up  into  factors,  then,  although  relations  may  exist  among  the  coef- 
ficients c,  they  are  not  of  such  a  nature  as  to  produce  any  simplifi- 
cation in  the  form  of  the  solution.  From  this  point  of  view  the 
equation  1)  is  in  this  case  a  general  equation,  or,  according  to  Kron 
ecker,  it  has  no  affect. 


ELEMENTAKY    CASES- — GROUP    OF    AN    EQUATION RESOLVENTS.        173 

On  the  other  hand,  if  for  particular  values  of  the  coefficients  c 
the  Galois  resolvent  F{^)  breaks  up  into  irreducible  factors  with 
rational  coefficients 

6)  F{^)  =  F,{^){F,{^)...FAr), 

then  the  unsymmetric  functions  Ff{:),  which  in  the  case  of  fully 
independent  coefficients  are  irrational,  are  now  rationally  known. 
The  equation  1)  is  then  a  special  equation,  or  according  to  Kron- 
ecker,  an  affect  equation.  The  affect  of  an  equation  lies,  then,  in  the 
manner  in  which  the  Galois  resolvent  breaks  up  or,  again,  in  the 
fact  that,  as  a  result  of  particular  relations  among  the  coefficients 
c,,r  ,,  .  .  .  c,,  or  the  roots  ,Xi,  x.,,  .  .  .  x„,  certain  unsymmetric  irreduci- 
ble functions  F;{:)  are  rationally  known.  The  determination  of  that 
which  is  to  be  regarded  as  rationally  known  in  any  case  is  obviously 
of  the  greatest  importance.  As  a  result  of  any  change  in  this 
respect,  an  equation  may  gain  or  lose  an  affect. 

If  the  group  belonging  to  any  one  of  the  functions  F;{c)  is  G,;, 
then  every  function  belonging  to  Gi  is  rationally  known,  being  a 
rational  function  of  F,.  It  is  readily  seen  that  the  groups  Gi  all  coin- 
cide. For  if  the  subgroup  common  to  them  all  is  J\  then  the  ration- 
ally known  function  aF^  -{-  f^F.,  -{-...  -\-  /.  F^  belongs  to  /',  and  conse- 
quently every  function  belonging  to  /'is  rationally  known.  Accord- 
ingly the  factor  of  F,  which  proceeds  from  the  application  of  the 
substitutions  of  /'  alone  to  any  linear  factor  r  —  a^Xi^  —  a.^Xf,^... —  «„»?,•„ 
of  F,  is  itself  rationally  known.  This  is  inconsistent  with  the 
assumed  irreducibility  of  F,-^  unless  J'=  Gi^=  G.>=^  ...  =  G^. 

All  the  functions  F^ ,  F.,,  .  .  .  F^,  therefore  belong  to  the  same 
family,  and  this  is  characterized  by  a  certain  group  G  or  by  any 
function  '/* {x^,x.2, .  .  .  x^  belonging  to  G.  Every  function  belong- 
ing to  G  is  rationally  known  and  conversely  every  rationally  known 
function  belongs  to  G. 

Theorem   I.     Every  special  or  affect  equation  is  character- 
ized by  a  group  G,  or  by  a  single  relation  betiueen  the  roots 
(/^{x,,x,,...x,,)  =  0. 

The  group  G  is  called  the    Galois  group  of  the  equation.     Every 
equation  is  accordingly  completely  defined  by  the  system 

->A  =  ^\ ,     ->Ai^V  =  C2,  .  .  . ;     ^ (a?, ,  ^2 ,  .  .  .  x,)  =  0.  * 

*C/.  Kronecker;  Grundziige  einer  arithmetischen  Theorie  der  algebraisclien  Gros- 
ser!, §§  10, 11. 


174  THEORY    OF     SUBSTITUTIONS. 

For  example,  given  a  quadratic  equation 

the  corresponding  Galois  resolvent  is 

^'^  —  2(ai  -f-  a^c^^  -}-  («!  —  a^'^Cc^  +  \a^a^c^  —  0. 

In  general  the  latter  equation  is  irreducible,  a*id  the  quadratic  equa- 
tion has  no  affect.     But,  if  w^e  take 

2ci  —  771  -j-  ?i,     C2  —  m  ?i, 
the  equation  in  c  becomes 

(c  —  a^m  —  o.^ri){^ — a^n  —  a2m)=0=(l — a^X^  —  a^x^{^^ -  -  a^x^  —  ag^j) 

and  the  given  quadratic  equation  has  an  affect. 
Again,  if  c^ — Ci^  =  0,  we  have 

(c  —  a^C^  —  a^c^^-=  0=  (c  —  a^x^  —  a^x^{'^  —  a^X^  —  «2^i)- 
But  if  C.2  —  2ci^  =  0,  we  obtain 

.^^  — 2(a,  +  «2)c,.^  +  2(ar  +  a2V-0, 
and  this  equation  has  no  affect,  so  long  as  we  deal  only  with  real 
quantities.     If  however  we  regard  i=  s/  —  1  as  known,  the  equa- 
tion has  an  affect,  for  the  Galois  resolvent  then  becomes 

(- —  («i  +  «2)  Ci  +  («i  —  "2)  c,i)  ($—  («!  +  flg)  Ci  —  («i  —  «2)  c,i)  =  0. 

§  154.  It  is  clear  that  every  unsymmetric  equation  <p(xi,  ...  x„) 
=  0  between  the  roots  produces  an  affect.  On  the  other  hand  an 
equation  between  the  cofficients  ^^(Cj,  C2,  . .  .  c„)  =  0  produces  an 
affect  only  when,  as  a  result,  the  Galois  resolvent  breaks  up  into 
rational  factors.  This  is  always  the  case  if  ^'^  is  the  product  of  all 
the  conjugate  values  of  an  unsymmetric  function, 

for  it  follows  then  from  §  2  that  one  of  the  factors  of  the  product 
must  vanish.  Whether  this  condition  is  also  necessary,  is  a  ques- 
tion which  we  will  not  here  consider. 

A  special  equation  might  also  be  characterized  by  several  rela- 
tions between  the  roots  or  the  cofficients  or  both.  A  direct  consid- 
eration of  the  latter  cases  would  again  present  serious  difficulty. 
From  the  preceding  Section  we  recognize,  however,  that  whatever 
the  number  or  the  character  of  the  relations  may  be,  if  they  pro- 


ELEMENTARY    CASES — GROUP    OF    AN    EQUATION RESOLVENTS.         175 

duce   an  affect,    they   can   all   be   replaced   by  a   single  equation 

\      1  9        2  ?    •    •    •   *^n)  ^* 

§  155.  We  will  illustrate  the  ideas  and  definitions  introduced 
in  the  preceding  Sections  by  an  example. 

Suppose  that  all  the  roots  of  an  irreducible  equation  f(x)  =  0 
are  rational  functions  of  a  single  one  among  them 

The  two  equations 

f{x)  =  0,f[<p,{x)\=Q 

have  then  one  root,  a?i,  in  common,  and  since  f{x)  is  irreducible, 
all  the  other  roots  cca  >  ^3 »  •  •  •  ^«  of  tlie  first  equation  are  also  roots 
of  the  second.     Consequently  /(a?)  =  0  is  satisfied  by 

<Px.{x2),  ^\{x^\  .  .  .  <Pk{x„), 
and  in  general  by  every 

^aM^i)]     (a,/?=2,3,...n). 
Again 

for  otherwise  the  two  equations 

/(«^)=0,       cp^{x)  —  cp^{x)  =  0 

would  have  the  root  <Py{x^  and  consequently  all  the  roots  of  f{x)  =  0 
in  common.  It  would  then  follow  that  ^^^(^i)  =  ^/sC^i)?  h  e.,  two 
of  the  roots  of  f{x)  would  be  equal.  This  being  contrary  to 
assumption,  it  appears  that  the  series 

Rl)  Xy,        <P2(Xy),    <Ps{Xy)     .     .     .     <Pn{Xy) 

coincides,  apart  from  the  order  of  the  elements,  with  the  series 

R2)  X,,     <f,{x,),  <p^{Xi),   .  .   .  <Pn{Xi). 

We  determine  now  what  substitutions  can  be  performed  among 
the  roots  x^,  X2,  .  .  .  x,^  without  disturbing  the  relations  existing 
among  them.  If  such  a  substitution  replaces  Xi  by  Xy  then  it 
must  also  replace  every  Xa  by  <pa(Xy).  The  substitution  is  there- 
fore fully  determined  by  the  single  sequence  x^jXy.  There  are 
accordingly  only  n  substitutions  which  satisfy  the  conditions.  These 
form  a  group  ^  (§  129);  for  the  system  is  transitive  and  its  degree 
and  order  are  both  equal  to  n. 


176  THEORY    OF    SUBSTITUTIONS. 

This  group  ii  is  the  group  of  the  given  equation.  For  the  rela- 
tions which  characterize  the  given  equations  are  equivalent  to  the 
single  relation 

^  ^  f^i  Ir,  —  <f,{x,)}  + ,%  [x,  —  ip,(x,)]  +  .  .  .  +  iK  [^.  —  <fM\)}  =  0, 
and  if  this    function  ^  is  to  remain  unaltered,  then  when  .r,   is 
replaced  by  Xy,  every  Xa  must  be  replaced  by  <fa(ory)  =  V^aC^yC^i)], 
exactly  as  under  the  application  of  the  substitutions  of  ii . 

§  156.  Without  entering  further  into  the  theory  of  the  group 
of  an  equation  we  can  still  give  here  two  of  the  most  important 
theorems. 

Theorem  II.  The  group  of  an  irreducible  equation  is 
transitive.  Conversely,  if  the  group  of  ari  equation  is  transitive, 
the  equation  is  irreducible. 

Thus  if  the  group  G  of  the  equation 

f{x)^(x—Xi)  (x  —  x.,)  .  .  .  {x  —  xj  =  0 

is  intransitive,  suppose  that  it  connects  only  the  elements  x^  ,x.,,...  Xa 
with  one  another.     Then  the  function 

<p  =  (x Xi)  (X X.j)  .  .  .  {X Xa) 

is  unchanged  by  G,  and  consequently  belongs  either  to  the  family 
of  G  or  to  one  of  its  sub- families.  In  either  case  <f  is  rationally 
known,  that  is,  the  coefficients  of  <f  are  rational  functions  of  known 
quantities,  so  that  c{x)  is  a  rational  factor  of  fix). 

Conversely,  if  f(x)  is  reducible,  a  factor  <p(x)  of  the  above 
form  will  be  rationally  known,  and  G  can  contain  no  substitution 
which  replaces  any  element  Xi,  x.,,  .  .  .  x^  by  ^a^i,  for  otherwise 
the  rationally  known  function  cr  would  not  remain  unchanged  for 
all  the  substitutions  of  G.     Consequently  G  is  intransitive. 

Theorem  III.  If  all  the  roots  of  an  irreducible  equation 
are  rational  functions  of  any  one  among  them,  the  order  of  the 
group  of  the  equation  is  n.  Conversely,  if  the  group  of  an  equa- 
tion is  transitive,  and  if  its  order  and  degree  are  equal,  then  all 
the  roots  of  the  equation  are  rational  functions  of  any  07ie  among 
them. 

The  first  part  of  the  theorem  follows  at  once  from  §  155.  We 
proceed  to  prove  the  second  part. 


ELEMENTARY    CASES GROUP    OF    AN    EQUATION RESOLVENTS.  177 

From  the  transitivity  of  the  group  follows  the  irreducibility  of 
the  equation. 

If  we  specialize  the  given  equation  by  adjoining  to  it  the  family 
belonging  to  x^ ,  the  group  will  be  correspondingly  reduced.  It 
will  in  fact  then  contain  only  substitutions  which  leave  x^  un- 
changed. But  as  the  group  is  of  the  type  ^i  (§  129),  it  contains  only 
one  substitution,  identity,  which  leave  x^'  unchanged.  Accordingly 
after  the  adjunction  of  x^ ,  all  functions  belonging  to  the  group  1 
or  to  any  larger  group  are  rationally  known.  In  particular 
Xi,  X.2,  .  .  .  Xn  are  rationally  known,  i.  e.,  they  are  rational  functions 
of  x^. 

From  this  follows  again  the  theorem  which  has  already  been 
proved  in  §  155: 

Theorem  VI.  If  all  the  roots  of  an  irreducible  equation 
are  rational  functions  of  any  one  among  them,  they  are  rational 
functions  of  every  one  among  them. 

§  157.  From  Th*eorem  III,  the  group  of  the  Galois  resolvent 
equation  of  a  general  equation  is  of  order  n!  To  obtain  it,  we 
apply  to  the  values 

all  the  substitutions  of  x, ,  x., ,  .  .  .  x„,  and  regard  the  resulting  rear- 
rangements as  substitutions  of  the  I's.  Since  every  substitution  of 
the  x's  affects  all  of  the  I's,  the  group  of  the  r's  belongs  to  the 
groups  ^ .  The  group  of  the  x^s  and  that  of  the  r's  are  simply 
isomorphic  (§  72). 

For  an  example  we  may  take  again  the  case  of  the  equation  of 
the  third  degree.  The  groups  G  ot  f{x)  —  0  and  F  of  F  (c)  =  0  are 
then 

Cr  =^  l^i,  (X-yX^),  \XiX^),  \X2X^),  \X-^X2X^),  yX^X^pC^) \ 

^  ^^  L-^'  (^1^3)  (^2^4)  (^5^6))  (^I'e)  (^2^5)  (^3'*)?  (^i"'2)  (^3^5)  (^4"b)» 

(^l^i^s)  ('»2^3-«6))  (^1^5"»4)  (^2^6^3)1- 

If  however,  the  given  equation  is  an  affect  equation  with  a 
group  G  of  order  r,  then  of  the  n !  substitutions  among  the  c's  only 
those  are  to  be  retained  which  connect  any  c,  with  those  I's  which 
together  with  c,  belong  to  one  of  the  rational  irreducible  factory 
F,{e)ot  F{^). 
12 


178  THEORY    OF    SUBSTITUTIONS. 

§  158.  We  apply  the  name  resolvent  generally  to  every  ;o-val- 
ued  function  <s(x^,  X2,  .  .  .  x„)  of  the  roots  of  a  given  equation 
f{x)  =  0.  The  equation  of  the  f^^  degree  which  is  satisfied  by  <p 
and  its  conjugate  values  is  called  a  resolvent  equation.  This  desig- 
nation is  appropriate,  in  that  the  solution  of  a  resolvent  equation 
■reduces  the  problem  of  the  solution  of  the  given  equation.  Thus, 
for  example,  in  the  case  of  the  equation  of  the  fourth  degree,  we 
employed  the  following  system  of  resolvents  (§  150): 

1)  The    2- valued  function    V  -^  =  {x^—x^  {a?i  —  0^3)  .  .  .{x^  —  x^, 

2)  The    6 -valued  function         (f  =  {x^x^  +  x^x^  +  ^  {x\X^  -\-  x^x^ 

^w\x^x^-\rx^x^\ 

3)  The  12- valued  function         c''  =  {x^x^ — x^x^  {x^x^-^-  x.jX^^ 

4)  The  24-valued  function         /  =  o-i{x-^ — x^  -\-  a^{x^ — x^. 

Originally  the  group  of  the  equation  was  of  order  24.  After 
the  solution  of  the  quadratic  equation  of  which  the  two-valued 
function  \f  J  was  a  root,  the  general,  symmetric  group  reduced  to 
the  alternating  group  of  12  substitutions.  The  extraction  of 
a  cube  root  led  then  to  the  group  G  (§150)  of  order  4;  another 
square  root  to  the  group  H  of  order  2;  and  finally  we  arrived  at 
the  group  1,  and  the  solution  of  the  equation  was  complete,  the 
function  -  being  superfluous. 

The  above  reduction  of  the  group  of  an  equation  to  its  //^^^  part 
by  the  solution  of  a  resolvent  equation  of  degree  f>  is  exceptional. 
For  general  equations  and  resolvents  this  reduction  is  not  possible. 
We  shall  see  later  that  it  is  possible  for  the  biquadratic  equation 
and  the  particular  series  of  resolvents  employed  above  only  because 
the  family  of  every  resolvent  was  a  self- conjugate  subfamily  of 
that  of  the  succeeding  one. 

§  159.  Given  any  /^-valued  resolvent  (p{x^,  x.>^  .  .  .  ir„),  this  sat- 
isfies an  equation  of  the  //^  degree 

To  determine  the  group  of  this  equation  we  adopt  the  same  method 
as  in  the  preceding  Section.     Suppose  that 

are  the  //-values  of  cr.     Every  substitution  of  the  group  G  of  th« 


ELEMENTARY    CASES GROUP    OF    AN    EQUATION RESOLVENTS.         179 

equation  f{-)c)  =  0  produces  a  corresponding  substitution  of  these 
fj  values.  The  latter  substitutions  form  the  required  group  of  the 
resolvent  equation  for  (f.  This  group  is  isomorphic  to  G,  and  it  is 
transitive,  since  G  contains  substitutions  which  replace  cpj  by  every 
(Px.  If,  in  particular,  the  group  of  <Pi  is  a  self -conjugate  subgroup 
of  G,  then  it  is  also  the  group  of  <P2,  <P^,  •  ■  -  <Pp'  The  latter  values 
are  therefore  all  rational  functions  of  v'l?  and  the  group  of  the 
resolvent  equation  is  a  group  ^,  as  in  §  156. 

§  160.  Following  the  example  of  Lagrange  *  we  might  attempt 
to  accomplish  the  reduction  and  possibly  the  solution  of  the  general 
Galois  resolvent  equation  F(^)  —  0  by  employing  particular  resolv- 
ents. It  will,  however,  appear  later  that  this  method  cannot  suc- 
ceed in  general  for  equations  of  a  degree  higher  than  the  fourth. 

*Mem.  d.  Berl.  Akad.  Ill;  and  Oeuvres  III.  p.  mr^  If. 


CHAPTER  X. 


THE  CYCLOTOMIC  EQUATIONS. 

§  161.  The  equation  which  is  satisfied  by  a  primitive  p^^  root 
of  unity  «>,  (p,  as  usual,  always  denoting  a  prime  number),  is  called 
the  cyclotomic  equation.     It  is  of  the  form 

1)  ''^^=x^-'-i-x^-'-j-  ...  ■i-x--\-x-irl  =  0, 
audits  p  —  1  roots  are 

2)  oj,aj\a}\..ioP-\ 

We  prove  that  the  left  member  of  1)  cannot  be  expressed  as  a  pro- 
duct of  two  integral  functions  <f(x)  and  (/'(x)  with  integral  coffi- 
cients.     For  if  this  were  the  case,  we  should  have  for  .r  =  1 

9(1)  ^''(1)  =  P. 
and  consequently  one  of  the  two  integral  factors,  for  example  <p{l), 
must  be  equal  to  ±1.     Moreover,  since  0{x)  =  O  has  at  least  one 
root  in  common  with  1),  at  least  one  of  the  expression  <p  (w*)  must 
vanish.     Consequently 

<p{aj^)  <p{w^^)  cp{u}^^)  .  .  .  C5(w/  -  ^  )  =:  0 

where  «>i  may  be  any  root  of  1),  since  the  series  2)  is  identical  with 
the  series  ui^^M^^u)^^  .  .  .  u)^^~^.     The  equation 

<p{x)  <p{x')  <p{x') . . .  (pix^-')  =  0 

has  therefore  all  the  quantities  2)  as  roots.  Consequently  the  left 
member  of  this  equation  is  divisible  by  1).     Suppose  that 

3)  <p{x)  <p(x') .  . .  <p{x^-')=F{x)  {xP-'+x^-'+  ...  +0^4- 1), 
where  F{x)  is  an  integral  function  with  integral  coefficients.    From 
8)  we  have  for  x  =  l 

Mi)y-'=pF(i), 

and  therefore  [tp(l)]^~\  which  is  equal  to  1,  must  be  divisible  by  p. 
Accordingly  1)  is  not  reducible. 


THE    CYOLOTOMIC    EQUATIONS.  181 

Theorem  I.     The  cyclotomic  equation  for  the  prime  num- 
ber p 

x^  —  1 


x—1 
is  irreducible. 


~x^-'-^x^-'-\-  ...  -{-x-\-l 


§  162.     If  now  g  is  a  primitive  root  (mod.  p),  then  the  series  2) 
of  the  roots  of  the  cyclotomic  equation  can  be  written 

Since  I)  is  irreducible,  the  corresponding  group  is  transitive.  There 
is  therefore  a  substitution  present  which  replaces  n)'-'  by  <y^l  Then 
every  w^*  is  replaced  by 

and  the  substitution  is  therefore 

The  jp  — 1  powers  of  s  form  the  group  of  1).  For  they  all  occur  in 
this  group,  and  from  §  156  the  group  contains  only  p  —  1  substitu- 
tions in  all. 

We  form  now  the  cyclical  resolvent 

in  which  a  denotes  a  primitive  root  of  the  equation 

5;^-^  — 1  =  0. 
For  brevity  we  write  with  Jacobi 

oi^J^  aa)^'  +  a?w9"-  -f  .  .  .  -f  a^-''a>s'^~^  —  (a,  u)). 

From  §  129  the  resolvent  (a,  u))^~^  is  unchanged  by  s  and  its  pow- 
ers, that  is,  by  the  group  of  the  equation.  It  can  therefore  be 
expressed  as  a  rational  function  of  a  and  the  coefficients  of  1). 

If  we  denote  a  {p — 1)*^  root  of  this  rationally  known  quantity 
by  Tj  we  have 

5)  (a,  w)  =  ri. 

The  quantity  r^  is  a  {p  —  1)- valued  function  of  the  roots  of  1).  It 
is  changed  by  every  substitution  of  the  group,  for  the  substitution 
s  converts  it  into 

(a,  w*')  =  a~^(a,  w)  =  a~"'ri. 


182  THEORY    OF    SUBSTITUTIONS. 

It  follows  from  the  general  theory  of  groups  that  every  function 
of  the  roots  can  be  rationally  expressed  in  terms  of  r^.  We  will 
however  give  a  special  investigation  for  this  particular  case.  The 
group  of  the  cyclotomic  equation  leaves  the  value  of 

6)  {a\w){a,wy-'-^ 

unchanged;  for  the  effect  of  the  substitution  8  is  to  convert  this 
function  into 

i.  e.,  into  itself.  If  now  we  denote  the  rationally  known  value  6)  by 
Ta,  where  in  particular  Tj^  "^  =  Tj,  we  obtain  for  /  =  1,  2,  .  .  .p  —  2, 
the  following  series  of  equations : 

(«,  <„)  =  r„    («^  ,o)  =  ^r,%      (./,  <«)  =  ^  r,^  .  .  .  {a--\  o.)  =^rf-\ 

Combining  with  these  the  obvious  relation  among  the  roots  and  the 
coefficients  of  1) 

(1,  «.)  =  -!, 

we  obtain  by  proper  linear  combinations 

It  is  evident  that  a  change  in  the  choice  of  the  particular  root  a 

p — 1 

or  of  the  particular  value  of  Tj  =  V  T^  only  interch«,nges  the  val- 
ue lo  among  themselves. 

Theorem  II.  The  solution  of  the  cyclotomic  equation  for 
the  prime  number  p  requires  only  the  determination  of  a  primitive 
root  of  the  equation  z^~^  — 1  =  0,  <^^<i  the  extractiofi  of  the 
(p  —  1)**  root  of  an  expression  which  is  then  rationally  knoivn. 
The  cyclotomic  equation  therefore  reduces  to  two  binomial  equa- 
tions of  degree  p  —  1. 


THE    CYCLOTOMIC    EQUATIONS.  183 

§  163.  The  second  of  these  operations  can  be  still  further  sim- 1 
plified.     The  quantity  T^  is  in  general  complex  and  of  the  form 

Ti  =  p{cosi'^-\-isini^). 

Since  now  (a,  a})^-^  and  («"~\  w~^)p  "^  are  conjugate  values,  it  fol- 
lows that 

{a,ioy-\a-\(jD-^)^-^  =zp  {cos  ^'>  +  *  s*^  '^ )  P  (cos  '9 — isin  '^)  =  /)l 
Again  it  can  be  shown,  exactly  as  in  the  preceding  Section,  that 

(a,aj){a-\oj-') 

belongs  to  the  group  of  the  cyclotomic  equation  and  is  conse- 
quently a  rational  function  of  a  and  of  the  coefficients  of  1).  If 
we  denote  its  value  by  U  we  have 

Accordingly  for  any  integral  value  of  k 

(s aj)=^/u [cos -jzrr + * ^*^  p_i  J 

Since  U  and  ''>  are  both  known,  we  have  then 

Theorem  III.  The  solution  of  the  cyclotomic  equation 
requires  the  determination  of  a  primitive  root  of  the  equation 
z^~^  — 1=0,  the  division  into  p  —  1  equal  parts  of  an  angle  which 
is  then  knotvn,  and  the  extraction  of  the  square  I'oot  of  a  known 
quantity. 

The  latter  quantity,  i/,  is  readily  calculated.     We  have 

To  reduce  this  product  we  begin  by  multiplying  each  pair  of  cor- 
responding terms  of  the  two  parentheses  together.     The  result  is 

Again,  if  we  multiply  every  term  of  the  first  parenthesis  by  the 
fc*^  term  to  the  right  of  the  corresponding  term  in  the  second  par- 
enthesis, we  obtain 

K)  a-^^(^-^^-  +  i_|_^-^^-  +  '+?_^.,,,-p^  +  2  +  <,2^        .y 

Now  w~^*+^  is  a  p*^  root  of  unity  m^  different  from  1;  for  if 


184  THEORY    OF    SUBSTITUTIONS. 

then  — gr*+1^0,  gr*E^l    (mod.  p),    i.    e.,    fc  =  0   or  p  —  1.     The 
quantity  K)  is  therefore  equal  to 

and  consequently 

?7  =  p—l  — («-'  +  «-'+  .  .  ,  -j-  a-^  +  ')=p  —  l  —  {—l) 
=  P 

Theorem  IV.  The  quantity  of  Theorem  III,  of  which  the 
square  root  is  to  be  extracted,  has  the  value  p. 

§164.  The  resolvent  5)  was  {p  —  1)- valued,  and  consequently 
the  preceding  method  furnished  at  once  the  complete  solution  of 
the  cyclotomic  equation.  By  the  aid  of  resolvents  with  smaller 
numbers  of  values,  the  solution  of  the  equation  can  be  divided  into 
its  simplest  component  operations. 

Suppose  that  p,  is  a  prime  factor  of  p  —  1,  and  that  p — 1—Piqi. 
We  form  then  the  resolvent 

where  «i  is  a  primitive  root  of  the  equation 

^^1  —  1  =  0. 
The  values  a^ ,  a^^,  .  .  .  a/\  are  all  dilBPerent,  and  the  higher  powers  of 
«i  take  the  same  values  again.     It  follows  that,  if 


l_|.^«^2^1-l_j_^p3^1-l_^   ^   ^   ^    ~j-<0»^l 


9,-1 


the  resolvent  above  can  be  written 

{fo  +  «1^1  +  «lV2  +   .  .  .  +  «l''^  ~^'^pi-  iV', 

or,  again  in  Jacobi's  notation, 

By  the  same  method  as  before  we  can  show  that  this  resolvent 
is  unchanged  if  to  is  replaced  by  w^,  that  is,  that  it  belongs  to  the 
group  of  1)  and  is  consequently  a  rational  function  of  a^  and  of  the 


THE    CYCLOTOMIC    EQUATIONS.  185 

coefficients  of  1).     We  denote  its    value   by   iVj  —  v/'i,    and   have 
accordingly 

If  then  we  write  precisely  as  before, 

it  appears  that  N\  is  rationally  known,  and  that 


These  several  functions  are  all  unchanged  if  co  is  replaced  by  w"^', 
that  is,  they  are  unchanged  by  the  subgroup 

.s^\  s-'^'i,  s'^\  .  .  .  .S'^^'i. 
We  have  therefore 

Tlieoreiii  V.  The  p^-valued  resolvents  s^o,  ^i ,  .  .  .  <^j,i-i  of 
the  cyclotomic  equation  belong  to  the  group  formed  by  the  powers 
of  s^'i.  They  can  be  obtained  by  determining  a  primitive  root 
of  z^\  —  1  =  0  and  extracting  the  p^''  root  of  a  quantity  which  is 
then  rationally  known.  * 

If  p.2  is  a  second  prime  factor  of  p  —  1,  and  if  p — -i  =2^iP2Q2i 
then  the  resolvent 

in  which  o..^  is  a  primitive  p/*^  root  of  unity,  is  reducible  to  the  form 

where 


This  resolvent  is  unchanged  if  oj  is  replaced  by  o)^^\  that  is,  it 
is   unchanged  by   the  substitutions    s^'i,  S"^i  .  .».       Consequently    it 
12a 


186 


THEORY    OF    SUBSTITUTIONS. 


can  be  rationally  expressed  in  terms  of  ^o?  if  «2  is  regarded  as 
known.     A^ain  if  we  write 

then  Ma  is  also  a  rational  function  of  <Pq^  and  we  have 


Theorem  VI.  T/ie  p^p^-valued  resolvents  /05/1)  •  •  •/pii»2-i 
of  the  cyclotomic  equation  belong  to  the  group  formed  by  the  poiv- 
ers  of  8^1^-'.  They  can  be  obtained  by  determining  a  primitive  root 
of  2: ^'2  — 1  =  0  and  extracting  a  p^'  root  of  a  quantity  which  is 
rational  hi  this  primitive  root  and  in  cr,,. 

§  165.     By  continuing  the  same  process  we  have  finally 

Theorem  VII,  If  p  —  1  =PiP2P.3  •  •  •  ?  the  solution  of  the 
cyclotomic  equation  for  the  prime  number  p  requires  the  determina- 
tion of  a  primitive  root  of  each  of  the  equations 

^J^i  —  l^rO,    ^^'2  — 1  =  0,    .^''•^    -1  =  0,  .  .. 

and  the  extraction  successively  of  the  p/'',  2h*^',  pi^^  ■  •  •  roots  of 
expressions  each  of  which  is  rationally  known  in  terms  of  the  pre- 
ceding known  quantities. 

If  9^0  is  given,  f  1,  ^2>  •  •  •  9'j>] —  1  are  rationally  known,  since  they 
all  belong  to  the  same  group.     Similarly  the  coefficients  of 

r     {X  —  O^)  {x—io^''^)  {x  —  io'P^)  ,  .  .  {x  —  co^^^--'^''^)  =  0, 
7)  J      {x  —  io9){x  —  a>^'^^')  (cc  — 0.^2^1  +  1)  .  .  .  (^_,,.(^x-i)^.+i)  3=0  , 

are  all  rationally  known.  Accordingly  after  the  process  of  Theo- 
rem V  has  brfen  carried  out,  the  equation  1)  breaks  up  into  p^  fac- 
tors 7).  Since  the  group  belonging  to  each  of  these  new  equations 
is  transitive  in  the  corresponding  roots,  the  factors  7)  are  again  all 
irreducible,  so  long  as  only  the  coefficients  of  1)  and  <Pq  are  known. 


THE    CYCLOTOMIC    EQUATIONS.  187 

Again,  after  the  process  of  Theorem  VI  has  been  carried  out, 
/o  is  known.  All  the  values  of  this  function  belong  to  the  same 
group  and  they  are  therefore  all  rational  functions  of  /,)•  Similarly 
the  coefficients  of 

are  rationally  known.  The  equations  7)  are  therefore  now  reduci- 
ble, and  each  of  them  resolves  into  P2  factors  8),  which  are  again 
irreducible  within  the  domain  defined  by  /(,.  We  can  proceed  in 
this  way  until  we  arrive  at  equations  of  the  first  degree. 

§  166.  The  particular  case  for  which  the  prime  factors  of  p  —  1 
are  all  equal  to  2  is  of  especial  interest. 

Theorem  VI II.  If  2"*  -{-1  is  a  prim^  number  p,  the  cyclo- 
tomic  equation  belonging  to  p  can  be  solved  by  means  of  a  series 
of  m  quadratic  equations.  In  this  case  the  regular  polygon  of 
p  —  2'"  -{- 1  sides  can  be  constructed  by  means  of  ruler  and  compass. 

In  fact,  for  one  root  of  the  cyclotomic  equation  we  have 

27:    ..    .    2r.         .     ,  27:        .    .     2t. 

uj  —  cos \-isin  — ,    to    '=:cos- ism  — , 

p  p  '  p  p' 

io-\-  oi    '  =  2  cos  — , 

p 

and  consequently  the  angle  —  can  be  constructed  with  ruler  and 
compass. 

In  order  that  2*^  +  1  may  be  a  prime  number,  it  is  necessary  that 
m  =  2'*.  For  if  m  =  2'*?^, ,  where  m^  is  odd,  then  2"*  + 1  =  (2^")'"'+ 1 
would  be  divisible  by  2""  +  1.     If 

/.-O,  1,2,3,4, 
the  values  of  p  are  actually  prime  numbers 
i>  =  3,  5,  17,  257,  65537, 

and  in  these  cases  the  corresponding  regular  polygons  can  there- 
fore be  constructed.      But  for  /j.  =  5  we  have 

2^  + 1  =  4294967297  =  641 .  6700417, 


188  THEORY    OF    SUBSTITUTIONS. 


.,^ 


SO  that  it  remains  uncertain  whether  the  form  2-4-1  furnishes  an 
infinite  series  of  prime  numbers.* 

§  167.  We  add  the  actual  geometrical  constructions  for  the 
cases  p  =  5  and  p  =  17. 

For  p  =  5,  we  take  for  a  primitive  root  g—2,  and  obtain  accord- 
ingly 

g'^-l,     ^^-2,     ^^  =  4,     ^^  =  3     (mod.  5). 

Consequently 

(T^—lU-f-  io\  Cy  =  (0^  -f  co^ 

— 1+\/5  — 1— V5 

fo  = 2 '    ^^  = 2—- 

If  oj-  is  substituted  for  w,  the  values  ctq  and  c>i  are  interchanged. 
The  algebraic  sign  of  V^  is  therefore  undetermined  unless  a  par- 
ticular choice  of  <o  is  made.     If  we  prescribe  that 

2;:    ,    .    .    2:r     • 
to  =  cos-^  -f- 1  Sin  -p- , 

then 

2-  ,        .  4- 

<ff,  —  co-{-oj*  =  2 cos-^  ,     (fi-=  11)-  -\-  o/  —  2 cos  -^ , 

consequently  <ff,  >  0,  c?i  <  0,  and  the  V5   in  the  expression  above  is 
to  be  taken  positive.     Furthermore 

/.o  =  ^,    Zi  =  ^*;    X >  =  '''',  )^3  =  <^^ 
—  1+ V5  +  *VlO  +  2V^ 

/o  =  «>  = ^ , 


,     — 1-h  V5-iVlO  +  2\/5 
/i=^  = ^ 


*  cy.'Gauss:  Disquisit.  arithni.,  §  362.  The  statement  there  made  that  Ferinat  sup- 
posed all  the  numbers  22'*  +1  to  he  prime,  is  corrected  by  Baltzer:  Crelle  87,  p.  172.  At 
present  the  following  exceptional  cases  are  known : 

2^^^  +1    divisible  by    641 (Landry), 

ol2 

2     +1    divisible  by    114689 (J.  Pervouchine), 

2^^  + 1    divisible  by    167772161. . .    (J.  Pervouchine;  E,  Lucas), 
2*^  +  1    divisible  by    274877906945    (P.  Seelhoff  ), 
C/.  P.  Seelhoff:  Schlomilch  Zeitschrift,  XXXI.  pp.  172-4, 


THE    CYCLOTOMIC    EQUATIONS.  189 

the  sign  of  i  being  so  taken  that  the  imaginary  part  of  oj  is  positive 
legative. 

For  the  construction  of 
the  regular  pentagon  it  is 
sufficient  to  know  the  re- 

solvent  c?o  =  2  gos-^  . 

^  ^  Suppose  a  circle  of  radi- 
us 1  to  be  described  about 
O  as  a  center.  On  the  tan- 
gent at  the  extremity  of 
the  horizontal  radius  OA 
a  distance  AE  =  lAO  —  i 
is  laid  off.     Then  . 


0E=  \/l+i 


If  now  we  take  EF=  E  O,  we  have 


/\/5  —  1 
AF  =  EO--EA  =  -^^ ^<Po, 

AF=2cos-^. 
5 

Finally  if  AF  is  bisected  in  G  and  GHJ  drawn  //'  to  OA  and 

5 


OC       to    HJ,    then    HOC^-COJ^~,   since  cosHOC  =  AG 


2tc 

=  cos-^.     H,  C,  and  J  are  therefore  three  successsive  vertices  of  a 

regular  pentagon. 

§  168.     For  p  =  17  we  take  for  a  primitive  root  g  =  6.     Accord- 
ingly we  have 

9',  g\  9\  g\  g\  q\  g\   g\  4.  g\  g'\  g'\  &\  g'\  g'\  g'\  g"; 

1,   6,    2,  12,  4,    7,    8,   14,  16,    11,  15,     5,    13,  10,     9,     3,    1; 

(p,=iO    +W^    -]-a>'   -{-lO^    -f  c«l«-(-o.l5_|_^13_^^9^ 
^^  =:,  ^6  _|_  ^12  _|.  ^7  _|_  ^U  _^  ^11  _|_  ^5  _^   ^iO  _^  ^3. 


190  THEORY    OF    SUBSTITUTIONS. 

To  find  cTq  V^i  ?  we  multiply  every  term  of  s^o  into  the  k^^  term  to 
the  right  of  that  immediately  below  it  in  ^j,  taking  successively 
A;  =  0, 1,  ...     We  obtain  then 

"Po  <Pi  =  <Pi-\-<Po  +  <Po+  9(>  -\-  ^1+  <Pi  +  <Pi  -\-  <Po  =  "^{n-^  <Pi)  =  —  4. 
Consequently 

r  +  <P—4:  =  o, 

_— 1-f  V17      _-i— vr? 

where  the  sign  of  V  17  is  undetermined  until  the  particular  root 
(o  is  specified.     If  we  take 

2::    ,     .    .    2;: 
to—  cos^-\-istnY=, 

we  have  for  the  determination  of  the  sign 

=  2  [^cosp=  -f-  cos-^-i-  cos^f  +  ^^^37"  J 

=  2  I^COSp^  — COS  ji^  — COS  j7^  —  COS  j^J  ^ 

and  V 17  above  must  be  taken  positive. 
We  have  further 

x,=io  +c.*+c.^«+c.^^       xi  =  -^'  +<^'  +-^'+«>^ 

;^,  =  a>«  +  «>^  +  a>"  +  «>^«,  ;^3  :=  a>i2_^  «,h  +  ..^  -f  «,3. 

;^0/l=/3+/l+/0+/2=   —1,        /2/3=/0  +  /3+Z2+/l=— 1; 


/o,/i=  2  =*"  Y — 4""'  /2,/3-  2  =^Y — 4 — • 

The  algebraic  signs  are  again  easily  determined.     We  have 


THE    CYCLOTOMIC    EQUATIONS.  191 

2r 


;,„  =  (<„+  <„»)  +  (««  +  ,o»)  =  2  (cos  ^  +  COS  1^)  >  0, 
X,  =  («,»  +<u")  +  (<.'  +  a,">)  =  2  (cos  ^  +  COS^)  <  0, 

Oonsequently 


With  /o  as  a  basis  we  proceed  further: 


■f-  ^S  =  ^05         V''oV\  =  /3  =  ^  +  a/^  +  1; 


^'o,  V'l  = 

9_  Q 

Since  now  v''o  = '^  cos -^ ,  v''i  ^2cos-^,  we  have  0o  >  ^^?  and  there- 
fore 


These  results  suffice  for  the  construction  of  the  regular  polygon 
of  17  sides.  Suppose  a  circle  of  radius  1  to  be  described  about  O 
as  a  center,  and  a  tangent  to  be  drawn  at  the  extremity  of  the  hori- 
zontal radius  OA.  On  the  tangent  take  a  length  AE  =  iOA  =  ^', 
then 

Vi7 


OE=\/l-{-j\=      ^ 
Further,  if  EF=EF'  =  E  O,  we  have 

V    ,j,_vi7-i_^o     ,^._vn+i_    ^1. 

AF-         -        _^,      AF-         ^         --2"' 


192 


THEOEY    OF    SUBSTITUTIONS. 


OF=^l^  +  l,  OF'  =  ^/^^t 


—M 


Taking  then 
we  have 


FH^FO,     F'H'  =  F'0, 


AH=AF-{-FO  =  ^  +  J^+l  =  Xo: 


Air  =  —  AF'  +  FO 


fi 


I=;f3. 


We  bisect  A  if  in  F;  then 


AY=ix,. 


THE    CYOLOTOMIC    EQUATIONS.  198 

We  take  now  AS^l,  and  describe  a  semicircle  on  H'S  as  a  diam- 
eter; if  this  meets  the  continuation  of  OA  in  K,  then 

AK^  =  AS.AH'=x^- 

Again  if  we  take  LK=AY  and  KL  =  LM  =  LN,  and  describe  a 
circle  of  radius  L  K  about  jL,  we  obtain 

AN-\-AM^2KL  =  2AY  =  Xo  =  4'^  +  ^'i 
AN.AM=:  AK^  =  AS.AH'=x^  =  M\. 

The  greater  of  the  two  lengths  A  2V,  A3f  is  equal  to  c^o;  we  may 
write  therefore 

2- 
AM=:0o  =  2cos-^. 

•    * 

If  P  is  the  middle  point  of  AM,  and  if  ^P  ;'  AG,  and  OD  , 

A  O,  then  Q  and  D  are  two  successive  vertices  of  the  required  reg- 
ular polygon  of  17  sides. 

§  169.     We  consider  now  the  case  Pi=2,  under  the  assumption 
p>2.     If  gr  is  a  primitive  root,  then 


ip^=  o>  -fwfl^+^/>^-j-  .  . 

.+-^"       , 

<p^  =:  w"  -f-  to^'  -|-  w^'  -f-  .  . 

.+-''^-', 

?'o  +  ^l  =  — 1- 

To  determine  ^o^i  we  use  the  same  method  as  in  the  preceding 
Sections,  and  obtain 

+    ... 

The  exponents  which  occur  here  in  any  bracket 

^^-+^  + 1,  gV^'  + 1),  9\g'^-^'-^  1), . . . 

are  plainly  either  all  quadratic  remainders,  or  all  quadratic  non- 
remainders,  or  all  equal  to  0.  In  the  first  case  the  value  of  the 
corresponding  bracket  is  c'o,  in  the  second  case  c^i,  and  in  the  third 

p  — 1 
case  — ^  . 

13 


194  THSJbEY    OF    SUBSTITUTIOI^S. 

Consequently 


S) 


I  1       P — 1 

<fo  <Px  —  Wi  <^o  +  >>^2  <-"]  4-  m^  -^  -  , 

?/ii  -j-  wio  +  »^3  =  — s— » 


where  m^ ,  mo ,  m^  represent  the  number  of  brackets  of  the  several 
kinds. 

If  gr'^+^  +  l^O  (mod.  i?),  then  2(2a  +  l)=p  — 1  and  accord - 

p 1 

ingly  — ^—  is  an  odd  number.     The  third  case  occurs  therefore  only 

when  p  =  4Ar-f-3,  and   then  only  once,  !'*>.  f or  a  =  — - — .      Con- 

sequently  m.  =  0  or  m-^  —  X  according  as  is  even  or  odd. 

Since  cr„  ctj  is  rational  aad  integral  in  the  coefficients  of  the 
cyclotomic  equation  1),  this  product  is  an  integer,  and  we  may 
therefore  take 

c'„  if,  —  m, — ^    —  n  =  —  n{  ^„  -f  <r,  ) 

where  n  is  an  integer.     It  follows  then  from  S)  that 

{m, -f  n ) cr,  +  (m,  +  w) ^^  —  0 

In  this  equation  all  the  powers  of  oj  can  be  reduced  to  powers  lower 
than  the  p^^\  and  the  equation  can  then  be  divided  by  lo.  The 
resulting  equation  is  then  of  degree  p  —  2  at  the  highest,  and  still 
has  the  root  uj  in  common  with  the  equation  1)  of  degree  p  —  1. 
It  is  therefore  an  identity  and 

m^  =  7112  =  —  ^^ 
Consequently  we  have  for  the  values  of  m^ ,  m^  and  cr^ ,  <p^ 


p — 1  p  —  1 


(^even) 
(V^odd) 


p  — 3  p  —  l     p— B_-pH-1     fp  —  l 

1— (-l)^p 
n<p^  =  — ^—^ — , 


THE    CYCLOTOMIC    EQUATIONS.  195 

l_(_l)-2^ 


i<P—<Po)  (?  —  9i)  =  <P^-\-<P 


4 


(--1)   '  P        ,  _-l-\(-l)   '  P 
^0-  2  '       ^'~  2  ' 

where  the  algebraic  sigQ  of  the  square  root  is  necessarily  undeterm- 
ined so  long  as  the  particular  value  of  w  is  not  specified. 

§  170.     We  consider  now  the  two  equations 

Z,^(x~o>^){x-~o)9^)  {x  —  io^')  .  .  .  (iC  — o>^^"-)  =  0. 

The  roots  and  consequently  the  coefficients  of  these  equations  are  un- 
changed if  (D  is  replaced  by  (o^\  If  therefore,  when  the  several 
factors  are  multiplied  together,  any  coefficient  contains  a  term 
mw«  it  must  also  contain  terms  m^^*^',  m.(o°-^\  .  .  . ,  that  is,  it  con- 
tains (Pq  or  ^1 ,  according  as  «  is  a  quadratic  remainder  or  non- 
remainder  (mod.  p).  Accordingly  every  coefficient  will  be  of  the 
form 


m!  if,,  +  m"  i?,  =  — ! 2~^ ' 

and  on  introducing  these  values  we  have 


__X-{-Y\(-l)'\ 


where  X  and  Y  are  integral  functions  of  x  with  integral  coefficients. 
Again  since  z^  is  obtained  from  z^  by  replacing  co  by  at^,  or  %  by  <Pi , 
that  is,  by  a  change  of  the  sign  of  the  square  root,  we  have 


Hence 


ZnZi 


.  _x-yM(-i 

■)'P 

2 

_x''—l      X'—{- 

-1)   '  pY' 

x—1  " 

2 

196  THEORY    OF   SUBSTITOTIOlfS. 

and  we  have 

Theorem  IX.     For  every  prime  number  p 

OP        JL  -^ 

wJiere  X  and  Y  are  integral  functions  of  x  with  integrctl  coeffi- 
cients. * 

*  The  extensive  literature  belonging  to  this  Chapter  is  found  in  Baehmann :  Lehre 
von  der  Kreistheilung,  Leipzig,  Teuhner,  1872.  The  present  treatment  follows  in  part 
the  method  there  employed.    The  two  figures  are  taken  from  Bachmann's  work. 


CHAPTER   XI. 


THE  abelia:n^  equations. 

§  171.  The  cyclotomic  equation  has  the  property  that  it  is  irre- 
ducible and  that  every  one  of  its  roots  is  a  rational  function  of  every 
other  one.  We  turn  now  to  the  treatment  of  the  more  general  prob- 
lem of  those  irreducible  equations  of  which  one  root  x\  is  a  rational 
function  of  another  root  x^.  Among  these  equations  the  cyclo- 
tomic equation  is  evidently  included  as  a  special  case. 

Suppose  that 

1)  /(a;)  =  0 

is  the  given  irreducible  equation,  and  that  two  of  the  roots  x\  and  x^ 
are  connected  by  the  relation 

2)  x\=::e{Xi), 

where  ^  is  a  rational  function.     Then 

SO  that  the  irreducible  equation  1)  has  the  root  Xi ,  and  consequently 
all  its  roots,  in  common  with  the  equation 

3)  f[e(x)]  =  0. 

In  particular  x\  —  0  (aji)  must  be  a  root  of  3),  so  that 

Consequently  0\d{xi)]  is  a  root  of   1)  and  therefore  of  3).     Then 
0{0[6(x^'\  \  is  a  root  of  1),  and  so  on.     With  the  notation 

e[e{x)-\^g'{x)-\,    e[ei'{x)-\  =  no{o^)'\  =  ^{^l 

it  appears  that  all  the  members  of  the  infinite  series 

X,,  e{x,\  e\x,\  e\x,\  ...  ^^0,  ... 

are  roots  of  the  equation  1).      Since  however  the  latter  has  only  a 
finite  number  of  roots  it  follows  from  a  familiar  process  of  reason* 


198  THEORY   OF     SUBSTITUTIONS. 

ing  that  there  must  be  in  the  series  a  function  0'^(xi)  which  is  equal 
to  the  initial  value  x^ ,  while  all  the  preceding  functions 

x„0(x,),0'\x,\...O"^-\x,) 

are  different  from  one  another.  The  continuation  of  the  series  then 
reproduces  these  same  values  in  the  same  order,  so  that  only 

0"*(X^)  z=z  0''^"{Xi)  =  .  .  . 

take  the  initial  value  Xi ,  and  that  for  k  <m  only 

0-^+\x,)  =  O''^+\x,)=  ... 

are  equal  to  0^{x-^. 

If  the  system  of  m  roots  thus  obtained  does  not  include  all  the 
roots  of  the  equation  1),  suppose  that  X2  is  any  remaining  root. 
Then  x^  also  satisfies  3),  and  therefore  0(x.^  is  a  root  of  1),  and  so 
on.     Accordingly  we  have  now  a  new  system  of  >i  different  roots 

x^,  0{x^\  0\x^\  . .  .0>'-\xo). 

But  since  the  equations 

4)  0-^{y)—y  =  O   0f'(z)  —  Z  =  O 

have  each  one  root  y=^Xi,z  =  X2  in  common  with  the  irreducible 
equation  1),  they  are  satisfied  by  all  the  roots  of  the  latter.  The 
former  equation  of  4)  is  therefore  satisfied  by  a?2,  the  latter  by  a?,, 
consequently  m  is  a  multiple  of  /^  and  ,0.  is  a  multiple  of  m,  i.  e. 
m  —  [i. 

Again  all  the  roots  of  the  second  series  are  different  from  those 
of  the  first.     For  if 

0\x,):=0^{x,)     (a,b<m), 

then,  on  applying  the  operation  6'"*-*,  we  should  have 

0-(x,)  =  x,  =  0-^-'(x,), 

and  a?2  would  be  contained  in  the  first  series,  which  is  contrary  to 
assumption. 

If  there  is  another  root  0^3,  not  included  among  the  2m  already 
found,  the  same  reasoning  applies  again.     We  have  therefore 

Theorem  !•  If  one  root  of  an  irreducible  equation  f{x)  =  0 
is  a  rational  function  of  another,  then  the  roots  can  be  divided  into 
V  systems  of  m  roots  eachy  as  in  the  following  table : 


5) 


O^HE    ABELIAN    equations.  190 

x„  o{x,),  o'\x.^^...e'""'{x,), 


X,,  d{x^\  0\x^),...0»'-\x,). 
The  function  d  is  such  that  for  every  root  Xa 

0^^X0)^X0., 
and  the  equation  f{x)  ~()  is  of  degree  mv. 

§  172.  '  We  can  now  determine  the  group  of  the  equation  1). 
Since  the  equation  is  irreducible,  its  group  is  transitive  (§  156);  it 
therefore  contains  at  least  one  substitution  which  replaces  x^  by  any 
arbitrary  x^ .  It  follows  then  that  all  the  roots  of  the  first  line  of 
5)  are  replaced  by  those  of  the  a*^^  line.  The  group  of  1)  is  there- 
fore non-primitive  and  has  >  systems  of  non-primitivity  of  m  ele- 
ments each.  The  number  of  admissible  permutations  of  the  v  systems 
is  not  as  yet  determinate;  in  the  most  general  case  there  are  v!  of 
them.  If  any  .r^  is  replaced  by  0^(xa),  then  every  0''(xa)  is  replaced 
by  O^'  +  ^ixa);  there  are  therefore  m  possible  substitutions  within  the 
single  system.  The  order  of  the  group  of  1)  is  therefore  a  multiple 
of  m''  and  a  divisor  of  vim*'. 

Theorem  II.  The  group  of  the  equation  1)  is  non-primitive. 
It  contains  >  systems  of  non-primitivity,  which  correspond  to  the 
several  lines  of  5).     The  order  of  the  group  is 

r=^r^  m^, 
where  r^  is  a  divisor  of  )^\. 

§  173.  In  the  following  treatment  we  employ  again  the  notation 
of  Jacobi 

where  w  is  a  root  of  the  equation  w*** — 1  =  0.  Similarly  we  write 
We  form  then  the  following  resolvents: 

^1  is  symmetric  in  the  elements  of  the  first  system  and  is  changed 
in  value  only  when  the  first  system  is  replaced  by  another;  it  is 
therefore  a  v-valued  function,  its  values  being   s^i,  ^2?  •  •  •  S^*" 


200  THEORY    OF   SUBSTITXTTIOIfR. 

Every  symmetric  function  ot  the  ^'s  is  a  rational  function  of  the 
coefficients  of  1) .     The  quantities 


that  is,  the  coefficients  of  the  equation  of  the  v*^  degree 

of  which  <fi^<p2', '  '  '<Pv  are  the  roots,  are  therefore  rationally  known. 
Theorem  III.     The  resolvent 

(1, 6{x,) )  -  a-i  +  e{x,)  +  o\oc,)  +  . . .  +  e""-\x,) 

is  a  root  of  an  equation  of  degree  -^ ,  the  coefficients  of  ivhich  are 
ratimially   knoum    in   terms    of    the  coefficients   of    the    equation 

§  174.  The  equation  6)  has  no  affect  (§158),  unless  further 
relations  are  explicitly  assigned  among  the  roots  x-^ ,  x., ,  .  .  .  'X\, .  If 
however  any  root  <fa  of  6)  has  in  any  way  been  determined,  the 
values  of  the  corresponding  Xa,  ^(^a),  ■  •  •  can  be  obtained  algebrai- 
cally by  exactly  the  same  method  as  that  employed  in  the  preceding 
Chapter. 

Thus,  the  equation  ^)  of  which  the  roots  are  x,  0(x),  0'\x),  .  .  . 
O'^'^x)  is  irreducible.  Its  group  consists  of  the  powers  of  the  sub- 
stitution (ldO\.  .  0^-').     And  if  we  write 

where  (o  is  now  assumed  to  be  a  primitive  m*^  root  of  unity,  we 
have 

le{x)-\-aje%x)-{-  ...+<o'--'e^{x)y"  =  aj^le{x)-{-  a>e\x)+...  (o^-'e^{x)] 

=  {<o,e{x)r, 

that  is  Ti  is  unchanged  by  the  substitution  {100^ ..  .  0""-^).  Conse- 
quently Tj  is  a  rational  function  of  the  coefficients  of  0)  and  of 
the  primitive  m*^  root  of  unity  «>.  The  m}^  root  of  this  known 
quantity  T^  we  denote  by  r^ .     Again  if  we  write 

{w\d{x))(<o,0{x)r-^=T„ 

it  can  be  shown  by  the  same  method  that  we  have  already  frequently 


THE  ABELIAN  EQUATIONS.  201 

employed  that  1\  is  also  rational  in  w  and  the  coefficients  of  S). 
Taking  successively  /  ==  0,  1,  2,  .  .  .  m — 1,  and  combining  the  result- 
ing equations,  we  have  then 


T,    '     '         ^^    X     .     •  •  •     .  rp^ 


The  function  r^  being  m -valued,  x  also  admits  only  m  values,  and 
these  coincide  with  x ,  0{x),  0^(x) ,  .  .  .  .  If  any  other  m"^  root  of  T^ 
is  substituted  for  r^ ,  the  m  values  of   x  are  permuted  cyclically. 

Theorem  IV.  If  the  m  roots  of  an  equation  of  degree  m 
are 

x,,d{x,),d\x,),  ...0--\x,), 

where  0(x)  is  a  rational  function  for  which  0*"(xi)  =  x^ ,  then  the 
solution  of  the  equation  requires  only  the  determination  of  a  primi- 
tive root  of  z"" — 1  —  0  and  the  extraction  of  the  m*^  root  of  a 
known  quantity. 

Theorem  V,  If  one  root  of  an  equation  of  prime  degree  is 
a  rational  function  of  another  root,  the  equation  can  he  solved 
algebraically. 

For  in  this  case  we  have  mv—p  and  m >  1 ;  consequently 
m=p  and  v  =  1. 

§  175.  If  all  the  coefficients  of  f{x)  are  real,  the  process  of 
the  preceding  Section  admits  of  further  reduction.     The  quantity 

T^=i{a),  d{x)f'  =  p  {cos  T?  +  ^  sin  ^9  ) 
can  be  rationally  expressed  in  terms  of  ^  and  the  coefficients  of  /, 
Tlie  latter  being  real,  the  occurrence  of    i=  ^  —1  in  T^  is  due 
eat jjely  to  the  presence  of  (o .     Consequently. 

T/  =  {(o-\  d{x)Y  =  p{cos^'^  —  isin^),     T^T^' =  p\ 

s/l\f;'=^/p':^(<o,d(x)){<o-\0{x))=U, 
18a 


w 


202  THEORY    OF    SUBSTITOTIONS. 

where  U  is  again  rationally  known,  since  it  is  unchanged  by  the 
group  of  <p.     We  have  then. 

T^z^  js/  U\  COS \-i8in — 1. 

V  m  m     J 

Theorem  VI.  //  all  the  coefficients  of  f(x)  are  real,  the 
second  operation  of  Theorem  IV  can  be  replaced  by  the  extraction 
of  a  square  root  of  a  known  quantity  and  the  division  of  a  known 
angle  into  m  equal  parts, 

§  176.  If  the  m  of  the  Theorem  IV  is  a  compound  number,  the 
solution  can  be  divided  into  a  series  of  steps  by  the  aid  of  special 
resolvents.  If  m^m^  mi,  where  m^  is  any  arbitrary  factor  of  w,  we 
take 

<\  =  ^1 + ^"'  (^i) + o'*-^  (^i)  H- . . .  +  0^^'^  -  'y^^  (xi) , 

02  =  0{xi)  -{-0^1+'  (xi)  4-  ^2,«i + 1  (^^)  4- ...  4-  ^(-'j  - 1)  -1 + 1  (xi) , 

4'n.,  =  0'^^-'{xi)  +  d''^i-'{xi)-\-0'^-\xi)-}- . .  .+e-'i-i-'{xi), 
and  consider  the  resolvent 

[xi+«i%,)+«i'^'(^i)+ .  •  •  +ar-'0--\x,)iri 

in  which  a^  is  a  primitive  rrii^  root  of  unity.  This  resolvent  is 
equal  to 

If  Xi  is  replaced  by  0{Xi),  then  <Pu  <p2^  --  -  4'm,  are  cyclically  permu- 
ted, and  the  resolvent  is  unchanged.  It  can  therefore  be  ration- 
ally expressed  in  terms  of  a,  and  known  quantities.  We  denote 
this  expression  by  N^  and  its  mj^  root  by  Vj ,  so  that 

If  then,  as  before,  we  write 

(^1 + «iV2 + «i  Va + . . . )  (^1 + «i  ^2 +«iV8  +...r^'^=N^, 

it  appears  that  N\  is  rationally  known,  and  that 

/  III    ■^S      2    I    "^8      3    I 


THE^ABELIAN    EQUATIONS.  203 

Theorem  VII.  The  mi-valued  resolvent  <Pi  can  be  obtained 
by  determining  a  primitive  root  of  2""!  — 1  =  0  and  extracting  the 
m^^  root  of  a  known  quantity. 

By  continuing  the  same  process,  we  obtain 

Theorem  VIII.  If  the  roots  of  an  equation  of  the  m** 
degree  are 

X,,  0{x,),  0\x,\  .  .  .  0—\x,),      [0-{x,)  =  1] 

and  if  m^^m^m^m^ .  .  . ,  the  solution  of  the  equation  requires  only 
the  determination  of  a  primitive  root  of  each  of  the  equations 

^^-.-l_Q^    5;-2_l=:0,    ^'"B— 1=0,,.. 

and  the  successive  extraction  of  the  m^^,  m^^,  m^^,  .  .  .  roots  of 
expressions,  each  of  which  is  rationally  known  in  terms  of  the  pre- 
ceding results. 

§  177.  The  solution  can  also  be  accomplished  by  a  still  different 
method. 

Suppose  that  m  =  m^m-i .  .  .  m^=  m^ri^  =  m^n^  =  .  .  .  =m^7iu . 
Then  we  can  form  the  following  equations: 

f  g,{x)=:  0,  with  the  roots  x,,  0-'i{x,),  ^'"K^i)»  •  •  •  ^>^"i  ~ '^"'^  (^i), 
and  with  coefficients  which  are  rational  functions  of  a  resol- 
vent /i  =  a?i  +  0"'i  (cci)  +  •  •  •      /i  is  a  root  of  an  equation 
h^  (x)  =  0  of  degree  m^ . 

(  g,{x)=0,  with  the  roots  .^i ,  6^'"<ir,),  ^'"Kic,),  . . .  ^(^'^-^"^^(a?,), 
and  with  coefficients  which  are  rational  functions  of  a  resol- 
vent /2  =  ^1  +  ^'^'-(^0  +  •  .  •     /2  is  a  root  of    an  equation 
hoijx)  —  0  of  degree  m.2 . 


A) 


A,) 


A^) 


g^{x)  =  0,wiilOitheTootsx,,  O^'^'^{x,),0''"'-{x,),  .  .  .  ^C"- -')"*" (a;,), 
and  with  cofficients  which  are  rational  functions  of  a  resol- 
vent Xo>  =  ^i-\-  ^""^  (^i)  +  •  •  •     Z<o  is  a  root  of  an  equation 
s  h^  ix)  =  0  of  degree  m^, . 
If  now  we  select  m^,  ma,  . . .  m^,  so  that  they  are  prime  to  each 
other,  then  the  equations 

g^(x)=0,    g2(x)  =  0,...g^{x)  =  0 


204  THEOBT    OF    SUBSTITUTIONS. 

have  only  the  one  root  x^  common  to  any  two  of  them.  By  the 
method  of  the  greatest  common  divisor  this  root  can  be  rationally 
expressed  in  the  coefficients  of  fifi ,  S'2 »  •  •  •  S'w ,  that  is,  in  the  coeffi- 
cients of  f{x)  and  in  Xu X2,  -  -  ■  /o^- 

The  solution  of  f{x)  —  0  depends  therefore  on  the  determination 
of  one  root  of  each  of  the  equations 

fh(x)  =  0,    h,(x)  =  0,..,K{x)-=0 

of  degrees  m^^m^,  .  .  .  m^,  respectively.     If 

m  =  Pi^'ipg""  •  •  •  Pw"''^ 

where  Pi,P2,  .  /.'pw  ar©  the  different  prime  factors  of  m,  then  we 
are  to  take 

mi  —  p«i,  m2  =p%  .  .  .  m«  =Pa,**^. 

If  for  any  one  of  the  equations  h\{x)  —  0  the  exponent  a^  is 
greater  than  1,  then  recourse  must  be  had  to  the  earlier  method  of 
solution  to  determine  a  /a  • 

§  178.  In  illustration  of  this  type  of  equations  we  add  the  two 
following  examples.     In  the  one  case  we  take 

^  ^     n^  +  ^i     roo  +  d' 

where  a^  =  a,  (3^  =  /?,  y^  —  y^  (Jj  —  ^5  are  real  quantities.  We  assume 
that  ao — /S;'4=0;  otherwise  we  should  have 

The  functions  0'^{x) ,  (i^{x) ,  .  .  .  will  also  be  linear  in  x  with  real 
coefficients.     We  may  write 

■  ■•■«-stl;.  '■'•)=;iSt- -">=SS- 

Again 

and  a  comparison  of  the  two  expressions  for  6''"(a?)  shows  that 

«„,  =  «i««._i  +  n/^m-l  ,        /5m  =  /5i«m-]  +  \['^m-\, 


THE    ABELIAN    EQUATIONS.  205 

From  these  equations  we  obtain  at  once  the  characteristic  relation 

7)  aj,n  —  /^,.  r»»  =  {'h'\  ~  1^1  Tl)     («.«  -  I'^m  -  1  —P^m-xYm-l) 


By  dividing  every  coefficient  of  f^{x)  by  ^/ ad — j3y  or  by  \/  f^y — «^^i 
according  as  ad — jSy  is  ppsitive  or  negative,  we  can  arrange  that  for 
the  new  coefficients  of.  0(x),  and  consequently  for  those  of  every 
0^(x),  the  relation  shall  hold 

8)  ad  —  flr=:±l. 

We  determine  now  under  what  conditions  it  can  happen  that 

0'^{x)  =  X. 

The  values  of  x  which  are  unchanged  by  the  operation  0  satisfy  the 
equation 

_  ax-j-ft 

yx^  +  (d  —  a)x  —  /S  =  0. 
For  these  fixed  values  we  have  therefore,  according  as  ad  —  j3y  =  ±  1, 


and  consequently 

A)     We  assume  in  the  first  instance  that  x'  and  x"  are  distinct, 
that  is  that  iV-j-1.     We  have  then 

0(x) X'    _  j^3C  —  x'         O'^jx)  —  x'   _      -^x  —  x' 

0{x)  —  x"~^j^^^"     0''(x)  —  x"~~       x  —  x'"'" 
0"ix)—x'        ^r^x—x' 
0^'^{x)—x"  x  —  x"' 

The  necessary  and  sufficient  condition  that  6^*"(x)  =  a?  is  there- 
fore that 


206  THEORY    OF    SUBSTITUTIONS. 


..=<,„.[^'_^(f+.')V.]-.,. 

This  condition  can  be  satisfied  by  complex  or  by  real  values  of 
the  quantity  in  the  bracket.  In  the  former  case  the  upper  alge- 
braic sign  must  be  taken,  and  further 

so  that  we  may  write 

a-\-d 

N*^  =  (^cos  f — i  sin  ipf"*"  =  cos  2  m  ^  —  i  sin  2  m<p. 

kit 
Accordingly,  we  must  have  2m^  =  2k7t ,  ^  =  — ,  and 

where  k  is  any  integer  prime  to  m.  If  the  condition  9)  is  fulfilled, 
the  function  &'*'{x)  will  be  the  first  of  the  series  d{x),  0\oc),  ...  to 
take  the  initial  value  x. 

If  the  quantity  in  the  bracket  is  real,  it  must  be  either  +  1  or  —  1, 
since  one  of  its  powers  is  to  be  equal  to  1.  The  case  iV  =  -|-  1  is  to 
be  rejected,  since  then  cf'  =  x" .     The  case  iV  =  — 1  gives 

a+r5=:0,    a'^  +  /?^  =  0, 

and  O'^ix)  =  x^  which  agrees  with  the  condition  9),  since  m  —  2. 
B)     It  remains  to  consider  the  case  x'  =  x".     We  have  then 


C4^)"*'=«- 


The  lower  sign  must  be  taken,  and  accordingly 

(2a  IF  1)^^+2/9 


It  follows  that 


0'\x)  = 


2rx-\-:{2'^^l)' 


0\x)  = 


_  (3a  ip  2)x  +  3/5 


Syx-}-{SdT2)' 


^^^  ~  mrx-\-  [md  =F  (m—  1)]* 
If  now  0'"{x)  =  a?,  we  have 


THE  ABELIAN  EQUATIONS.  207 

yx^  -f-  (<5  —  a)x  —  /5  =  0, 

that  is,  we  must  have  already  had  0{x)  =  x.  And,  again,  it  is  clear 
that,  as  m  increases,  0"\x)  approaches  the  limiting  value 

^  ^     rx-\-{5^l) 
We  have  shown  therefore  that 

m 

where  k  is  prime  to  m,  are  the  sufficient  and  necessary  conditions 
that  d*^{x)  shall  be  the  first  of  the  functions  0\x)  which  takes  the 
initial  value  x.     For  ??i  =  2 ,  the  second  condition  is  not  required. 

§  179.  For  the  second  example  we  take  for  d{x)  any  integral 
rational  function  of  x  with  constant  coefficients. 

For  every  integral  m  the  difference  O'^'ix)  —  x  is  divisible  by 
d{x)  —  x.     For  if 

6{x) — x=^{x — >z^{x — z^  . ,  .  {x — 2;^), 

then  for  every  Za  0{Za)  =  Za,  and  consequently  d'^(Za)  =  ^^(Za)  =  . . . 
=  6\Za)  =  Za .   Moreover 

^^+i(aj)_^^(a;)  =  l0^(x)—z,-]  ld\x)—z,-]  . . .  l0\x)—z;], 
and  consequently 

d''  +  \x)—e\x)  _  d\x)—Z,  e\x)—Z^  0\Xy)  —  Z^  -T>  {    \ 

"  \X\         X  X  ~~^  Z-y  X        Z^  X        Zy 

where  P  is  a  rational  integral  function  of  x  and  of  the  coefficients 
of  0,  since  it  is  symmetric  in  the  roots  z-^^z^^  •  -  -Zy.  If  now  we  take 
k  successively  equal  to  0, 1,  2,  . . .  m — 1,  and  add  the  resulting 
equations,  we  have  as  asserted 

d-^{x)  —  x  =  [0{x) — £c]  Q{x\ 

where  ^  is  a  rational  integral  function  of  x.  From  this  equation  it 
follows  that  for  every  root  of 

Q{x)  =  Q 

we  have  0'^{x)  =  x,  and  conversely  that  every  root  of 


THEORY    OF    SUBSTITUTIONS. 

which  is  not  contained  among  z^,  z^,  .  .  .  Zt,  also  makes   Q{x)  vanish. 
Every  root  ^  of  Q{x)  =  0  therefore  gives 

and  consequently  also 

so  that  d{^),  and  likewise  0\^),  e\^\  ...  are  all  roots  of  Q{x)  =  {). 
Again  since  ^  is  different  from  the  z's,  0{^)  if,  and  0{^)  =Za. 

Theorem  IX.     If  6{x)  is  a  rational  integral  function  of  x 
of  degree  v ,  then  the  roots  of  the  equation  of  degree  (v  —  l)m 

0  [X) X 

can  be  arranged  as  in  Theorem  I,  5).     If  m  is  a  prime  number  then 
each  of  the  v  —  Irows  of  5)  contains  m  roots  * 

§  180.     Conversely  if  the  equation  f(x)  =  0  has  the  roots 
x^,  x^  =  d{x^\  x^  =  0\xo\  . . .  x^^i  =e^-\xy,     \0^{x^  ==  io] , 
every  one  of  these  roots  will  also  satisfy  the  equation 

^{x)-  x  =  0, 
but  no  one  of  them  will  satisfy 

e{x) — a?  =  0; 
consequently  /  {x)  is  a  divisor  of  the  quotient 

The  restriction  that  e(x)  shall  be  an  integral  function  is  unessen- 
tial.    For  if  d{x)  is  fractional 

where  gx  and  grg  are  integral  functions,  then  in  r 

Qi^  ^  ^  gi{^o)iQ2{x,)g^{x,) . . .  g^jx^-i)'] 
92(^0)  g2{xi)92(x,) . . .  g2(xm-i) 
the  denominator,   being    a  symmetric    function  of    the    roots  of 
/(a;)  =  0,  is  a  rational  function  of  the  coefficients  of  f{x);  and  the 


THE  ABELIAN  EQUATIONS.  209 

second  factor  of  the  numerator,  being  symmetric  in  x^,  x^,  .  .  .  x^_i 
is  a    rational   integral   function   of   Xq.      Consequently  0(xq)  is    a 
rational   integral   function   of  x^,   which  can   be   reduced   to    the 
(m  —  1)'^  degree  by  the  aid  of  /(xq)  =0. 
We  have  therefore 

Theorem  X.     Every  polynomial  of  the  equations  treated  in 
§  174  is  a  factor  of  an  expression 

or(x)  —  x    . 
d,(x)—x  ' 

where  0^{x)  is  an  integral  function  of  the  (m — Vf^ degree. 

For  example,  if  we  take  6^  — x^ -\-hx-\-c,  we  may  reduce  this 
by  the  linear  transformation  y  =  x-\-aio  the  form  O^^^x^-^-  a.    Then 

0,\x)  —  x  =  {0,{x)  —  x)  [x' 4-ic'  +  (3a+  l)a;*  +  {2a-\-l)x' 

+  (3a^  +  3a  +  l)a)^  +  (a' +  2a  +  l)a;  4- (a=^  +  2a^  +  a  +  1)  J- 
The  discriminant  of  the  second  factor  on  the  right  is 

jz=r_(4a  +  7)(16a2H-4a  +  7)l 
If  now  we  take 

■   4a  +  7=— fc^     a=  —  ^^^, 

the  second  factor  breaks  up  into  two,  and  this  is  the  only  way  in 
which  such  a  reduction  can  be  effected.     We  have  then 

6'i(a;)— a? 
[8x'  +  4(l  +  A;K— 2(9  — 2A;  +  A;>  — (1  +  7A:— A;'  +  A;^)] 
[8i«'4-4:(l  — fcK— 2(9  +  2A;  +  fc>— (1  — 7fc— A;^— A;^)] 

or,  for  A;  =  2x  +  1,  a  =  — /^ — /  —  2, 

0  ^(nr\ 'T 

^yz:^  =  [^^  +  (A  +  l)^^-(A^+2)^-(A^  +  A^  +  2A  +  l)] 

[a^'  — Aa^^  — (A2  +  2A  +  3)a^  +  (/l=*  +  2P  +  3/l-f  1)]. 

In  this  way  we  obtain  the  general  criterion  for  distinguishing  those 
equations  of  the  third  degree  the  roots  of  which  can  be  expressed 
by  £c,  0{x),  6^\x).  In  the  first  place  0  must  [be  reducible  to  the 
form 

d=:x'—{X'-{-X-\-2) 
14 


210 


THEORY    or     SUBSTITUTIONS. 


that  is,  6^  —  4c  must  be  of  the  form  4  (/^  + ;.  +  2)  ^  (2A  -f  1)2  -|_  7 . 
then  to  every  d  there  correspond  two  equations  of  the  required  type 

^'  +  (>^  +  lK-(A^  +  2)x  — (/^  +  /2  +  2/  +  l)==0 
x'—Xx"  -  (A2  +  2A  +  3)^  +  {X'  +  2A^  +  3>^  +  1)  =  0. 

It  appears  at  once,  however,  that  0  is  unchanged  if  X  is  replaced  by 
—  {X  4-  1),  and  that  the  first  equation  is  converted  into  the  second  by 
this  same  substitution.  It  is  sufficient  therefore  to  retain  only  one 
of  the  two  equations. 

§  181.     We  introduce  now  the  following 

Definition.  If  all  the  roots  of  an  equation  are  rational  f  mic- 
tions of  a  single  one  among  them, 

then,  if  these  rational  relatio7is  are  such  that  in  every  case 

OaO^ix,)  =  Op0,{x,), 

the  equation  is  called  an  ^^Ahelian  Equation^  * 

We  have  already  seen  (§  173)  that,  if  the  roots  of  an  equation  are 
defined  by  5),  the  resolvents 

cp,  =:  (1,  0,{X,)  ),    <p,  =  (1,  0,{X,)  ),...<p,=  (1,  0,{X,)  )• 

satisfy  an  equation  6)  of  degree  >  the  coefficients  of  which  are 
rationally  known.  We  noted  further  that  this  equation  is  solvable 
only  under  special  conditions.  These  conditions  are  realized  in  the 
present  case.     We  proceed  to  prove 

Theorem  XI.  Abelian  equations  are  solvable  algebraic- 
ally. ** 

In  the  first  place  we  observe  that  since  cr^  is  symmetric  in  x^ , 
^i{xi),  .  .  .  <?i'"~^(a7i),  every  symmetric  function  of  these  quantities  is 
rational  in  <Pi  and  the  coefficients  of  1).     If  now  we  consider 

<P2  =  0C2+  e,{x,)  +  0,'(x,)  +  . . .  +  0- -^x,), 

and  assume  that 

X2  =^  ^2(^1)) 
we  have 

<P2  =  Mx^)  +  0,0,(x,)  +  0,%{x,)  +  .  .  .  +  Or-'0,{x,) 

= u^i) + ^2^1(^1) + ^2^i'(^i)  + . . .  +  ^2^r-^(.^i) 

=  Slx„0,{x,),  d,'{x,), . . .  y-X^Q]  =  R{<P,). 

*  C.  Jordan :  Trait6  etc.    §  402. 
♦•Abel:  Oeuvres  completes,  I,  No.  XI;  p.  114-140. 


THE  ABELIAN  EQUATIONS.  .  211 

For  from  0^0.^{x)=  OoO^ix)  follows  also 

0;%{x)  =  OlO,0,(xy}  =  ^hO^  \Olx)\  =  0,Oi{x) ; .  .  . 
The  equation  (f.y  =  -R(^i)  shows  that  6)  has  in  the  present  case  also 
the  property  that  its  roots  are  all  rational  functions  of  a  single  one. 
We  write  now  accordingly 

Then 

M<Pl)  =  <Pa  =^a+  0{X^)  +  0\X^)  +   .  .   . 

-  Oa{x,)  +  0{0^(x,))  +  0%0^{X,))  +  .  .  .  , 

K'Wd  =  e^{o^(,ic,))  +  oo^{o^{x,))  +  o'o^{o^{.x,))  +  . . . 

-  0^{0^{x,))  +  0{0^0^{x,))  +  0\0^0,[x,))  +  .  .  . 

so  that  the  operations  '"/  are  again  commutative,  like  the  0''s.  The 
equation  6)  is  therefore  itself  an  Abelian  equation,  the  degree  of 
which  is  reduced  to  the  m^^^  part  of  that  of  1). 

We  can  then  proceed  further  in  the  same  way  until  we  arrive  at 
equations  of  the  type  treated  and  solved  in  §  174. 

§  182.  The  character  of  a  group  of  an  Abelian  equation  is 
readily  determined  as  follows: 

Suppose  that  any  two  substitutions  s  and  t  of  the  group  replace 
,an  arbitrary  element  a^i  by 

^2  =  ^^2(^1),         ^-3=^3(^1), 

respectively.     Then  st  and  ts  replace  x^  by 

oMx,).    o,o,{x,), 

respectively.     But  since  by  assumption  0^f).^[x^)  —  0^0^{x^\  it  follows 

that 

,  st  =  ts. 

All  the  substitutions  of  the  group  are  therefore  commutative. 

If  conversely  the  group  G  of  an  equation  f{x)  =  0  consists  of 
-commutative  substitutions,  we  consider  first  the  case  where  G  is 
intransitive  and  f{x)  is  accordingly  reducible  (§  156).  Suppose 
that         •  ■  . 

fix)=Mx)f,{x)... 

where  fi{x),f2{x),  .  .  .  are  rationally  known  irreducible  functions. 
If  we  consider  the  roots  of  fi{x)  —0  alone,  every  rationally  known 


212 


THEORY    OF    SUBSTITUTIONS. 


function  of  these  is  unchanged  by  the  group  G  and  conversely. 
Accordingly  we  obtain  the  group  G^  belonging  to  f^{x)  =  Q  by 
simply  dropping  from  all  the  substitutions  s^ ,  S2 ,  «3  ?  •  •  •  of  G  those 
elements  which  are  not  roots  of  f-^{x)  =0,  and  retaining  among  the 
resulting  substitutions  ''•j ,  ^2  >  '^s  >  •  •  •  those  which  are  distinct.  It  is 
clear  that  from  SaS^  =  s^Sa  follows  also  <ya<^B  =  <^^<^a-  The  group  of 
every  irreducible  factor  of  f{x)  is  therefore  itself  composed  of  com- 
mutative substitutions,  and  is  moreover  transitive. 

We  may  therefore  confine  ourselves  to  the  case  where  G  is  a 
transitive  group.  If  now  G  contains  a  substitution  s^ ,  which  leaves 
Xi  unchanged  and  replaces  X2  by  x^,  then  if  we  select  any  substitu- 
tution  S2  of  G  which  replaces  x^  by  X2 ,  we  have 

^1^2  —  (_'^i«^2  •  •  •    /j        ^2^1  —  yX-iX-^  .  .  .    J. 

This  being  inconsistent  with  the  commutative  property  of  the  group,, 
it  follows  that  every  substitution  of  G  either  affects  all  the  elements,, 
or  is  the  identical  substitution. 

If  now  any  arbitrary  root  x^  of  f{x)  =0  is  regarded  as  known,, 
the  group  of  the  equation  reduces  to  those  substitutions  which 
leave  x^  unchanged,  i.  e.,  to  the  identical  substitution.  Conse- 
quently every  function  of  the  roots  is  then  rationally  known;  in 
particular  X2,  x^,  .  .  .  are  rational  functions  of  x^ , 

•^2  ^^^  ^2 \ "^l  ) ?      X-i'^  0^(^X1 ),  .  .  . 
Again,  if  Sa  and  s^  replace  x^  hy  Xa  =  0^[xi)   and  x^  =  0^[x^y 
respectively,  then  SaS^  and  s^Sa  replaces  x^^  by  0^0 ^[x^)  and  0^0^[x^)y 
and,  as  Sa,s^  =  s^Sa ,  the  operations  0  are  also  commutative. 

Theorem  XIT.  The  substitutions  of  the  group  of  an  Abe- 
lian  equation  are  all  commutative.  Conversely,  if  the  substitutions 
of  the  group  of  an  equation  are  all  commutative,  the  irreducible 
factors  of  the  equation  are  all  Abelian  equations. 

§  183.  The  substitutions  of  the  group  of  an  Abelian  equation^ 
as  well  as  the  relations  between  the  roots,  therefore  fulfill  the  con- 
ditions assigned  in  the  investigations  of  §§  137-139. 

In  particular  we  can  arrange  the  roots  in  the  following  systenL 

6/'^02^^^-0,h  .  .  .  0,^"^{x,)      {h,  =  0,  1,  2,  .  .  .  77,-1 ), 
WjWgWs .  .  .  7if,=^n, 


THE  ABELIAN  EQUATIONS.  213 

in  which  every  root  occurs  once  and  only  once.  The  numbers 
Til ,  712 ,  .  .  .  n^fc  are  such  that  every  one  of  them  is  equal  to  or  is  con- 
tained in  the  preceding  one,  and  that  they  are  the  smallest  numbers 
for  which 

respectively.  There  is  only  one  substitution  of  the  group  of  the 
equation  which  converts  x^  into  ^a(^i).  Denoting  this  by  Sa,  we 
can  arrange  the  substitutions  of  the  group  also  in  a  system 

s,^sM'^ .  .  .  s,"^     {h,  =  0, 1,  2,  .  .  .  n,  ~1). 
niU^n^ .  .  .  nk  =  n, 

where  again  every  substitution  occurs  once  and  only  once,  and  cor- 
responding to  the  properties  of  the  ^'s, 

The  numbers  nj,  ^2,  .  .  .  n^-  are  the  same  as  those  for  the  e's. 
To  form  a  resolvent  we  take  now 

•  M^i)  =  2  ^'''^'"'  •  •  •  ^''^^'^     (/^,  =  0, 1,  .  .  .  n,  - 1) 
and  construct  the  cyclical  function 

/i(^i)  =  [</'t(^i)  +  ^i^ivS(^i)  -f  ^i'^i%''i(a^i)  +  . . .  +  <^-'o,"^-'H^i)']\ 

where  (o^  is  a  primitive  n,^^  root  of  unity.  Then  ;^i(a:^i)  is  unchanged 
by  the  group  of  the  equation.  For  the  substitutions  of  the  sub- 
group 

G.2=   |S2,S3,    ■   "Sj,\ 

leave  </'](cCi)  unchanged,  and  the  powers  of  s^  convert  4'i{xi)  into 

respectively,  so  that  these  do  not  affect  the  value  of  xi  •  Conse- 
quently Xi  is  rational  in  the  coefficients  of  the  given  Abelian  equa- 
tions and  in  m.  From  Theorem  IV,  4'i  is  therefore  a  root  of  a 
"simplest  Abelian  equation"  of  degree  n^.  With  v''i  all  the  func- 
tions which  belong  to  the  subgroup  G^  of  G  are  also  rationally 
known. 

Again,  if  we  take 

^'i .  2(^1) = 2  ^''^''  •  •  •  ^'''  (''»• = 0. 1 .  • .  •  ^.— 1). 

1>o,...hh 


214 


THEORY    OF    SUBSTITUTIONS. 


and  form  the  cyclical  resolvent 

71,2(^1)  = 

in  which  (o.,  is  a  primitive  yi^^^^  root  of  unity,  the  function  /j  .^(x^)  is 
unchanged  by  the  group  G2 ,  and  is  therefore  a  rational  function  of 
^''1.     For  the  substitutions  of  the  group 

leave  (/'^ ,  o  unchanged  and  the  powers  of  s.y  convert  v^'i ,  2  iuto 

%''l,2,        f^2V'],2,.   .    -^"-"^''1,2, 

respectively.  Applying  Theorem  IV  again,  we  obtain  ^'j  ^  2  from  (/\ 
by  the  solution  of  a  second  simplest  Abelian  equation  of  degree 

^2- 

In  general,  if  we  write 

V],2,...|/—      ^j'V  +  i  ...'a;      (,a^ij, 

/] , 2 ,  . . .  V  =^ 

[0l.2,...v+^Av^,2....v+^.V0^2....v^-...+^/•'-'^.''•'-v^.2....J"^ 

the  value  of  ^''1,2,...  v  is  determined  from  that  of  the  similar  func- 
tion 


...-.=2'''''"'---'''''(^')- 


by  the  aid  of   a   simplest  Abelian  equation,   as  defined  by  Theos 
rem  IV. 

By  a  continued  repetition  of  this  process  we  obtain  finally  * 

Theorem  XIII,     If  the  n  roots  of  an  Abelian  equation  are 
defined  by  the  system 

,ho,h  .  .  .  e.'^^ix,)     (/I,  ^  0,  1,  2,  .  .  .  n,—  1) 
n^n^n^.  .  .nj,  =  n^ 

the  solution  of  the  equation  can  be  effected  by  solving  successively 
k  '''' simplesV^  Abelian  equations  of  degrees 

ni,n2,ns,  .  .  .  n,,. 
*L.  Kronecker:  Berl.  Ber.,  Nachtrag  z.  Dezemberheft,  1877;  pp.  846-851. 


THE  ABELIAN  EQUATIONS.  215 

§  184  The  solution  of  irreducible  Abelian  equations  can  also 
be  accomplished  by  another  method,  to  which  we  now  turn  our 
attention. 

Theorem  XIV.  The  solution  of  an  irreducible  Abelian 
equation  of  degree  n  =  p^^'-^p^'^- .  .  . ,  where  Pi,P2-,  •  .  .  cife  the  differ- 
ent prime  factors  of  n,  can  be  reduced  to  that  of  k  irreducible  Abe- 
lian equations  of  degrees  Pi"i,  P2^,  .  .  . 

The  proof  *  is  based  on  the  consideration  of  the  properties  of 
the  group  of  the  equation.     For  simplicity  we  take  n  ^Pi'^ip^"-^ 

Since  the  order  of  the  group  is  r  =  n,  the  order  of  every  one  of 
the  substitutions  is  a  factor  of  n,  and  is  therefore  of  the  form 
Pi"ip/2.  Every  substitution  of  the  group  can  accordingly  be  con- 
structed by  a  combination  of  its  (^2"^)'^  power,  (which  is  of  order 
Pl^)  and  its  {p"^y^  power  (which  is  of  order  pa^O-  Consequently  we 
can  obtain  every  substitution  of  the  group  G  by  combining  all  the 

the  orders  of  which  are  a  power  of  Pi ,  with  all  the 
/"    /"    t"  i" 

^     1)''    2)*'     Z->   •   '   •  ^    r2) 

the  orders  of  which  are  a  multiple  of  p^ .     Since  the  f  s  are  all  com- 
mutative, the  substitutions  of  G  are,  then,  all  of  the  form 

s  =  {t'j'^t\...){r^r',ri,..). 

The  order  of  the  product  in  the  first  parenthesis  is  a  power  of  Px , 
and  therefore  a  factor  of  Pi*i.     For  we  have 

Two  substitutions 

{t'J'^'-'){i"^t"^'--\       {t'J\..,){t"at'\...) 
are  different  unless  the  corresponding  parentheses  are  equal  each  to 
each.     For  if  the  two  substitutions  are  equal,  we  have 

{rj'^  .  .  .)  {t'J\  ...)-'  =  {t",t'\  .  .  .)-'{f'at"e  .  .  .), 
and  since  the  order  of  the  left  hand  member  is  a  divisor  of  Pi\  and 
that  of  the  right  hand  member  a  divisor  of  p^^  each  of  these  divi- 
sors is  1. 

*  O.  Jordan ;  Trait6  etc.  §  405-407. 


216-  THEORY    OF    SUBSTITUTIONS. 

The  number  of  substitutions  s  is  equal  to  n  —  p^'^\p2°-^.  And 
since  the  substitutions  t'  form  a  group  and  every  substitution  of  this 
group  is  of  order  p{^\  the  order  of  the  group  itself  must  be  p{^^, 
(§  43).  Similarly  the  Order  of  the  group  formed  by  the  t"  's  is 
equal  to  p^..     It  follows  then  from 

n  =  Pi°-^P2^  =  p^^p.^'i' 
that  mi  =  aj ,  m<i^=^  a^. 

Suppose  now  that  ^  is  a  function  belonging  to  the  group  of 
the  t"  's.  Then  (f  has  pi*i  values,  and  is  the  root  of  an  equation 
of  degree  p^'^^  the  group  of  which  is  isormorphic  with  the  group 
of  the  t'  's.     This  is  therefore  an  Abelian  equation. 

Similarly  the  function  0  belonging  to  the  group  of  the  f  s  is  a 
root  of  an  Abelian  equation  of  degree  p^'^-. 

If  now  (f  and  (p  have  been  determined,  the  function 

belongs  to  the  group  1.  Every  function  of  the  roots,  and  in  particu- 
lar the  roots  themselves,  are  rational  functions  of  x  >  ^^^  the  theorem 
is  proved. 

§  185.  Theorem  XV.  The  solution  of  an  irreducible  Abe- 
lian equation  of  degree  p"  can  be  reduced  to  that  of  a  series  of 
Abelian  equations  the  groups  of  which  contain  only  substitutions  of 
order  p  and  the  identical  substitution. 

If  G  is  the  group  of  such  an  equation,  the  order  of  every  sub- 
stitution of  (r  is  a  power  of  p.  Suppose  that  p^  is  the  maximum 
order  of  the  substitutions  of  G.  Then  those  substitutions  which  are 
of  orders  not  exceeding  p^~^  form  a  subgroup  iJ  of  G.  For  if  t^ 
and  ^2  are  two  of  these  substitutions,  then  from  the  commutative 
property  it  follows  that  ^ 

so  that  t^t^  is  also  of  order  not  exceeding  p^~\ 

If  the  group  H  is  of  order  p%  any  function  (f  belonging  to  H 
will  take  p*"**  values  and  will  therefore  satisfy  an  equation  of 
degree  p*^""".  If  we  apply  to  <p  the  successive  powers  of  any  sub- 
stitution r  oi  G  which  does  not  occur  in  iJ,  <p  will  take  only  p  values, 
since  r^  is  contained  in  H.     The  substitutions  among  the  values  of  <p 


THE  ABELIAN  EQUATIONS.  217 

which  are  produced  by  the  substitutions  of  G,  and  which  form  a 
group  isomorphic  to  G,  are  therefore  all  of  order  p.  From  the  iso- 
morphism of  the  two  groups  it  follows,  as  in  the  preceding  Section, 
that  the  equation  for  cp  is  an  Abelian  equation. 

If  (f  is  known  the  group  G  of  the  given  Abelian  equation  reduces 
to  H.  We  denote  the  group  composed  of  those  substitutions  of  H 
which  are  of  order  p^"^  or  less  by  iJi.  If  (fi  is  a  function  belong- 
ing to  -ffj ,  then  ^1  is  determined  from  <p  by  an  Abelian  equation  of 
degree  p"~"i,  the  group  of  which  again  contains  only  substitutions 
of  order  p,  and  so  on. 

§  186.  Theorem  XVI.  The  solution  of  an  irreducible 
Abelian  equation  of  degree  p*,  the  group  of  which  contains  only 
substitutions  of  order  p  and  the  identical  substitution,  reduces  to 
that  of  a  irreducible  Abelian  equations  of  degree  p. 

Although  this  Theorem  is  contained  as  a  special  case  in  that 
obtained  in  §  183,  we  will  again  verify  it  by  the  aid  of  the  method 
last  employed. 

Let  Si  be  any  substitution  of  the  group  G  of  the  given  Abelian 
equation;  then  the  order  of  s^  is  p.  Again  if  S2  is  any  substitution 
of  Gnoi  contained  among  the  powers  of  Sj,  then  since  SjSa  =  S2S1,  the 
group  if  =  j  S] ,  So  [  contains  at  the  most  p^  substitutions.  It  will 
contain  exactly  this  number,  if  the  equality  s{'S2  —  Si%^  requires 
that  a  =  a ,  6  =:  y5.  But  if  SjV  =  ^1%^,  then  s^" -*  =  s/-^  and  for 
every  value  of  /^ — b  different  from  0  we  can  determine  a  number 

m  such  that 

m( /S  —  6)  ^  1      ( mod.  p). 
It  follows  then  that 


S2-(^-M  =  Si'»(' 


which  is  contrary  to  hypothesis.     Accordingly  /5  =  b  and  a  =  a. 

If  a  >  2,  suppose  that  s^  is  a  substitution  of  G  not  contained 
among  the  p^  substitution  Sj^Sa^.  Since  s^s^  =  s^s^  and  S2S3  =  s^s.2 , 
the  group  H^—  {si,S2,  s^\  contains  at  the  most  p^  substitutions. 
And  it  contains  exactly  this  number,  for  if  Si^s.^^s^"  =  s^'^s./s^y,  then 
Sg'y"';  ==:si"~"S2^~^,  and  so  on,  as  before. 

Proceeding  in  this  way,  we  perceive  that  all  the  substitutions  of 
G  can  be  written  in  the  form 

s,^^s,\  .  .  .  s/%     (A.-  =  0,  1,  ...  p  —  1) 


218  THEORY    OF    SUBSTITUTIONS. 

where  every  substitution  occurs  once  and  only  once  (c/.  §  183).     If 
now  we  take  for  the  resolvents  and  the  corresponding  groups 

^a        V  «^1  5  "^2  )    •  •   •  '^n)  '■)        -^   a         "^    ]  ^1  )  ^2  ?   •   •   •  ^a  —  1  f  ) 

S-'^a  —  1  (,  ^1  )  ^2  )    •   •   •   '^n)\        -"   a  —  1  ^^^   '(^1^  ^'2i   •   ■   •  ^'a  —  2  ?  '"^a  C  ) 


then  every,  resolvent  depends  on  an  Abelian  equation  of  degree  p^ 
The  roots  of  the  given  equation  of  degree  p'^  are  rational  functions. 

of  <fi,  <f2,  •  '  '  fa,  for  the  function 

"/'  =  1^1  ^1  +  /52^2  +  .  .  .  +  i^a<Pa 

belongs  to  the  group  l(c/.  §  177). 

§  187.     The  p*"  roots  of  such  an  equation  may  be  denoted  by 

^-,,-,...--a  i^^  =  ^,  1,  2,  .  .  .p—1) 

Suppose  that  ^^^ ,  ^„, . . .  ^^  i^  *^®  ^oot  by  which  5/182^2 .  .  .  sj°-  replaces 
a*,,  g^       ,  .     Then  the  substitution 

by  virtue  of  the  left  hand  form  will  replace  a?^j ,  s„ , . . .  z^  by  the  root 
by  which  sf^s.^^ ^^sj'^  replaces  ^^i,^„,...  ^a-  ^^^  from  the  right 
hand  form  this  root  is  ^^1+^1,^2  +  12 ...  ^a+^a-  Consequently  every 
substitution  Si^'  s/-'  .  .  .  sj"^  replaces  any  element  ^^, ,  ^o , . . .  ^a  ^J 

that  is  the  substitutions  of  the  group  are  defined  analytically  by  the 
formula 

Si'^s^y  ..  .s^-'^=\z,,z,.  .  .  Za,     Zi-{-k,,Z2-\-  ko,  .  ..Za-^ka\     (mod.  p). 

The  group  of  an  Abelian  equation  of  degree  jp**,  the  substitutions 
of  which  are  all  of  order  p,  consists  of  the  arithmetic  substitutions 
of  degree  p°-  (mod.  p). 

§  188.  Finally  we  efFect  the  transition  from  the  investigations 
of  the  present  Chapter  to  the  more  special  questions  of  the  preced- 
ing one. 

Let  n  be  any  arbitrary  integer  and  let  the  quotient  —  be  denoted 
by  a.      Then,  as  is  well  known,  the  n  quantities 


THE    ABELIAN    EQUATIONS.  21^ 

COS  a ,    COS  2a ,    cos  3a ,  .  .  .  cos  na 
satisfy  an  equation,  the  coefficients  of  wliich  are  rational  numbers^ 

C)  «,«_  l,^^"-^+  ,v  'i(^l=^);«"-^-  .  .  .  =:0. 

If  now  we  write  x  =  cosa,  then  for  every  integer  m 

cosm.a  =  0(^cosa), 

where  ^  is  a  rational  integral  function.  Similarly  if  the  value 
cosniia  is  denoted  by  di(cosa),  we  obtain,  by  replacing  a  by  Wja  in 
the  last  equation,  the  result 

cos{'mniia)  =  b{cos7n^a)  =  66^  (cos a) 

Again  if  in  the  equatioL  Qi(cosa)  =  cos(m^a)  the  argument  a  is 
replaced  by  ma,  the  result  is 

cos  {rrii  ma)  =  6^1  (cos  ma)  =  6^0  cos  a. 

Consequently  the  roots  of  C)  are  so  connected  that  every  one  of 
them  is  a  rational  function  of  a  single  one  among  them,  x,  and  that 

e^d{x)  =  ddi  (x)     {x  =  cosa). 

The  equation  C)  is  therefore  an  Abelian  equation.    Accordingly 

2- 

X  =  cosa=  cos  — 

n 

can  be  algebraically  obtained.     We  have  here  an  example  of  §  181. 

§  189.     Suppose  now  that  n  is  an  odd  prime  number,  n  =  2^  +  1- 
Then  the  roots  of  the  equation  C)  are  the  following: 

Since  the  last  root  is  equal  to  1,  the  equation  C)  is  divisible  by  x — 1. 
The  other  roots  coincide  in  pairs 

2m-    _  cos  (2y-f-l — m)27r 

""""^^H^l  ~  2v  +  l  • 

Consequently  we  can  obtain  from  C)   an  equation  with  rational 

coefficients,  the  roots  of  which  will  be  the  following 

2rr         '              4:7:  2>7: 

COS^ — -,       COSrz — -  ,    .  .  .  cos. 


2,-1-1'     ^^"2v  +  l' 2v4-l 

This  equation  is  of  the  form 


220  THEORY    OF    SUBSTITUTIONS. 

-j- 5^^  ^       ^ U^  ...    —  U. 

With  the  notation 

2- 
^^^  o — T~T  =  cosa  =  X 

Zv-j-  1 

we  have  then 

2m- 
cos-^ — TT  ~  ^v^/  ~  cos  ma, 

so  that  the  equation  Ci)  has  also  the  roots 

e{x\  d\x),  d\x\  . . . 


that  is 


cos  a,     cos  m  a,     cos  in^  a,     cos  m^a,  .  ,  .  cos  m^  a,  .  .  . 


If  now  g  is  any  primitive  root   (mod.  2v  + 1)  then  the  v  terms  of 
the  series 

Ri)  cosa,     cosga,     cosg'^a,  .  .  .  cosg*'~^  a, 

are  distinct,     For  from  the  equation 

cos  g*^  a  =  cos  g^  a     («  >     ;  a,    <  v) 

it  would  follow  that 

g'^a=±g^a-{-2k7:, 

or,  replacing  a  by  its  value   „     '     , 

Zv  "p  JL 

g<^^:g^  =  g^(g<^-^zfl)  =  k{2v-\-l). 

Dividing  both  sides  of  this  equation   by  g^,    and  multiplying  by 
g,a-^  _j_  i^  ^Q  obtain  the  congruence 

gKa--^)^l     (mod.  2v  +  l). 

But,  since  since  2(a — /S)  <  2v,  this  congruence  is  impossible.     Con- 
sequently cosg'^a  is  different  from  cosg^a. 

Again 

cos  g^a  =  cos  a. 

For  since  f""'" — 1  =  {g" — l)(g*'+  1)  is  a  multiple  of  2v  +  l,  one 
of  the  two' factors  is  divisible  by  2v-|-l,  so  that 

^v=±l  +  fc(2n  +  l), 
and  consequently  the  relation  holds 


THE  ABELIAN  EQUATIONS.  221 

Gosg^a  —  cos[±  1 +  ^(2''  + l)]a  =  cos{±:  a-|-2fcr:)  =  cosa. 

It  follows  then  that  the  v  roots  of  the  equation  CJ  are  all  contained 
in  the  series  R^,  or  again  in  the  series 

X,  d{x\  b\x\  ...e^-'{x), 

while  e''{x)  =  1.  The  equation  Oj)  can  therefore  be  solved  algebrai- 
cally.    We  have  an  example  of  §  174. 

If  we  have  v  =  iii  n^  .  .  .  n^ ,  it  appears  that  we  can  divide  the  cir- 
cumference of  a  circle  in  2v-}-l  equal  parts  by  the  solution  of  to 
equations  of  degrees  Ui,  n2,  .  .  .  n^^.  It  n^,7i2,  .  .  .  n^  are  prime  to 
each  other,  the  coefficients  of  these  equations  are  rational  numbers. 
(§176). 

In  particular  if  v  =  2'^,  we  have  the  theorem  on  the  construction 
of  regular  polygons  by  the  aid  of  the  ruler  and  compass. 


CHAPTER   XII. 


EQUATIONS  WITH  IIATIONAL  RELATIONS  BETWEEN 
THREE  ROOTS. 

§  190.  The  method  employed  in  §  183  is  also  applicable  to  other 
cases.  We  will  suppose  for  example  that  all  the  substitutions  of  a 
transitive  group  G  are  obtained  by  combination  of  the  two  substitu- 
tions s,  and  80,  which  satisfy  the  conditions  1)  that  the  equation 
,s^«  =  s/  holds  only  when  both  sides  are  equal  to  identity,  and  2) 
that  SoSi  =  8i^s<,  (Cf.  §  37).  If,  then,  the  orders  of  Si  and  S2  are  Ui 
and  n.,,  all  the  substitutions  of  G  are  represented,  each  once  and 
only  once  by 

s/'is/"-'     (h,=  0,l,2,  ...n,  — 1). 

Suppose  now  that  G  is  the  group  of  an  equation  f{x)  —  0.  We 
construct  aresplvent  ^  =  4'o  belonging  to  the  group  1,  Si ,  s^,  .  .  .  s"^~\ 
and  denote  the  functions  which  proceed  from  9'-^  on  the  application 
of  S2,  So",  .  .  .  s/'2  "^  by  (J'l,  (/'2^  .  .  .  <r'v,o-i-  Then  all  these  9'''s  belong 
to  the  same  group  with  (/^o .     For  from  s.,Si  =  Si^s\,  we  have 

from  which  it  follows  at  once  that  the  powers  of  s^  form  a  self  con- 
jugate subgroup  of  G.     The  resolvent 

/  —   y'o  +  ''^2<r''l  +  «^2%\'  -f    ■    .   .   ''^2"-   "  Vno  -  1  ]  "^ 

is  therefore  unchanged  by  every  Sj",  and  since  Sg  permutes  the  ^'''s 
cyclically,  '/  remains  unchanged  by  all  the  substitutions  of  the  group, 
and  can  be  rationally  expressed  in  terms  of  the  coefficients  of  f{x). 
We  can  therefore  obtain  c'-^,  (/\,  ...  by  the  extraction  of  an  no^^  root, 
as  in  the  preceding  Chapter.  The  group  of  the  equation  then  re- 
duces to  the  powers  of  s^ ,  and  the  equation  itself  becomes  a  simplest 
Abelian  equation. 

§  191.  Again,  if  a  transitive  group  consists  of  combinations  of 
three  substitutions  Si,s.2,  s^,  for  which  1)  the  equations 


RATIONAL    RELATIONS    BETWEEN    THREE    ROOTS.  223 

are  satisfied  only  when  both  sides  ol  each  equation  are  equal  to 
identity,  and  2)  the  relations  hold 

then  all  the  substitutions  of  the  group  are  represented,  each  once 
and  only  once  by 

where  n^^n.,,  n^  are  the  orders  of  8^,82,8^.  If  now  G  is  the  group 
of  an  equation,  we  can  show  by  precisely  the  same  method  as  before, 
that  the  equation  can  be  solved  algebraically. 

Obviously  we  can  proceed  further  in  the  same  direction.  That 
groups  actually  arise  in  this  way  which  are  not  contained  among 
those  treated  in  the  last  Chapter  is  apparent  from  the  example  on 
p.  89,  where  8281=^81^-%. 

§  192.  Returning  to  the  example  of  §  190,  we  examine  more 
closely  the  group  there  given.  If  we  suppose  80  to  be  replaced  by 
its  reciprocal,  it  follows  from  the  second  condition  that  6\>~^SiS2  =  Sj^. 
From  Sy  we  can  therefore  obtain  every  possible  6\,  by  the  method  of 
§  4G.  We  have  only  to  write  under  every  cycle  of  s^  a  cycle  of  Sj^" 
of  the  same  order,  and  to  determine  the  substitution  which  replaces 
every  element  of  the  upper  line  by  the  element  immediately  below 
it.     This  substitution  will  be  one  of  the  possible  s/s. 

We  consider  separately  the  two  cases  1)  where  8^  consists  of  two 
or  more  cycles,  and  2)  where  8^  has  only  one  cycle. 

In  the  former  case  the  transitivity  of  the  group  is  secured  by  So. 
Consequently  every  cycle  of  S]^^  must  contain  some  elements  different 
from  those  of  the  cycle  of  Sj  under  which  it  is  written.  It  is  clear 
also  that  all  the  cycles  of  Sj  must  contain  the  same  number  of  ele- 
ments. Otherwise  the  elements  of  the  cycles  of  the  same  order 
would  furnish  a  system  of  intransitivity.  The  order  of  the  cycles 
can  then  obviously  be  so  taken  that  the  elements  of  the  second  cycle 
stand  under  those  of  the  first,  those  of  the  third  under  those  of  the 
second,  and  so  on,  so  that  with  a  proper  notation  the  following  order 
of  correspondence  is  obtained 


224  THEORY    OF    SUBSTITUTIONS. 

Si  =  {XiX.2       X^  .  .  .  )  (t/j^2       2/3  •..)•••. 

^1     —  (2/12/1 +i 2/1 +2A;  •   •  •  )  (^1^1 +4^^1+2*  •••)••• 

It  follows  then  that 

S2  =  (x,y,z,  .  .  .  ){x2yi  +  j,z,  +  j,2  ...)••• 

The  group  is  therefore  non- primitive,  the  systems  of  non-primitivity 
being  a^j ,  cr, ,  .  .  . ;  2/1 5  2/2  ?  •  •  • ;  ^  >  ^2  ?  •  •  •  The  substitutions  Sj*  leave 
the  several  systems  unchanged,  the  substitutions  Si%  permute  the 
systems  cyclically  one  step,  Si%^'  two  steps,  and  so  on.  Accordingly 
every  substitution  of  the  group  except  identity  afPects  every  element. 
The  group  is,  in  fact,  a  group  i2  (§  129). 

The  adjunction  of  any  arbitrary  element  x^  reduces  the  group  to 
the  identical  substitution.  Consequently  all  the  roots  are  rational 
functions  of  any  one  among  them. 

The  following  may  serve  as  an  example: 

Si  =  {x,x,x^)  (2/12/22/3),     s,  =  {x,y,)  {x,y.)  {x,y,), 
.    s,s,  =  (X1I/2)  {x,y,)  (xgi/g)  =  s,% . 

§  193.  In  the  second  case,  where  s^  consists  of  a  single  cycle, 
the  transitivity  is  already  secured.  We  may  write  then,  as  in  Chap- 
ter YIII, 

Si  =  \z      Z-\-l\,      s^'^=\z      Z-{-  a\ 

To  construct  the  s./s  we  proceed  as  before  and  obtain  from 
S]   =  (X1X2     a?3       '  '  '^^     (k~>  1) 

^1     —-^  \^i^i  -f  k^i  +  2k  •   •   •  ) 

the  series  of  substitutions 

Jc^ 1 

§2  =  1-^^     kz  +  i  —  k\,     s/=\z     k^z^—-—{i—k)\. 

Now,  in  the  first  place,  it  is  easily  shown  that  the  group  contains 
substitutions  different  from  identity,  which  do  not  affect  all  the  ele- 
ments. For  among  the  powers  of  s^  there  is  certainly  one  s^'*  which 
has  a  sequence  of  two  elements  in  common  with  S2.  Then  Sj'^Ss"^ 
does  not  affect  all  the  elements. 

Again,  it  can  be  shown  that  there  is  no  substitution  except  iden-^ 
tity  which  leaves  the  elements  unchanged.     For*  we  have 

hP 1 

s,'^s/  =  \z     mzJra)-{--j^—^ii-k), 


• 


RATIONAL    RELATIONS    BETWEEN    THREE    ROOTS.  225 

and  if    X\    and   Xx^i  were    not   affected  by  the    substitution   we 

should  have 

m 1 

fc?(;,  +  «)  +  |_i(i_fc)=;, 

^/3 1 

/fc^(^  + 1 +«)  +  ___  (;_fc)=/i  + 1, 
and  consequently 

The  substitution  then  becomes 

s,'^s/=z\z     z-\-a\, 

and  since  Xx  and  x^^i  are  unchanged,  a^O,  and  the  substitution 
is  identity. 

The  following  is  an  example  of  this  type: 

,  Sj  :=  \XiX<fC^X^XrpC^j  ,       §2  ^^  \X1X2)  yX^pC^)  (^4*^5/ 

From  the  preceding  considerations  we  deduce 

Theorem  I.  //  the  group  of  an  equation  is  of  the  kind 
defined  in  §  .190,  all  the  roots  of  the  equation  are  rational  functions 
of,  at  the  most,  tivo  among  them,  and  the  equation  is  solvable  alge- 
braically. 

§  194.  We  turn  now  to  the  converse  problem  and  consider 
those  irreducible  equations,  the  roots  of  which  are  rational  func- 
tions of  two  among  them: 

oo-i  =  f,{x, ,  Xn),  x^  =  <Pi(x, ,  X2),  ...x,=  ^n{x, ,  x,y. 

If  any  substitution  of  the  group  G  of  such  an  equation  leaves 
0*1  and  X2  unchanged,  it  must  leave  every  element  unchanged.. 
Again,  if  Sa  and  «'«  are  any  two  substitutions  of  G  which  have  the 
same  effect  on  both  x^  and  a?2,  then  s'aSa"^  leaves  x,  and  Xo  un- 
changed; consequently  s'^Sa^^^  =  1,  and  s'a  =  Sa. 

Suppose  now  that  the  substitutions  of  G  are 

There  are  n{n  —  1)  different  possible  ways  of  replacing  x^  and  X2 
from  the  n  elements  Xi,X2,  .  .  .  x^.  If  any  one  of  these  ways  is 
not  represented  in  the  line  above,  let  t^  be  any  substitution  which 
produces  the  new  arrangement.  Then  the  substitutions  of  the  line 
15 


226 


THEORY    OF    SUBSTITUTIONS. 


'2'^2  ?  ^2^*3  ?   •    •   •  f  2^'/- 

will  replace  x^  and  X2  by  pairs  of  elements  which  are  all  different 
from  one  another,  and  none  of  which  correspond  to  the  first  line. 
If  2r  is  still  <n{n  —  1),  then  there  are  other  pairs  of  elements 
which  do  not  correspond  to  either  line.  If  4  is  any  substitution 
which  replaces  Xi  and  X2  by  one  of  these  pairs,  we  can  construct  a 
third  line 

and  so  on  until  all  the  n{n  —  1)  possibilities  are  exhausted.  We 
have  therefore 

*  Theorem  II.  The  order  of  the  group  of  an  irreducible 
equation  of  the  n*^'  degree,  all  the  roots  of  which  are  rational  func- 
tions of  two  among  them,  is  a  divisor  of  n{n  —  1). 

§  195.  The  equations  of  the  preceding  Section  are  not  yet* 
identified,  however,  with  those  previously  considered  in  §  190. 
This  will  be  clear  from  an^example.  The  alternating  group  of  four 
elements  contains  no  substitution  except  identity  which  leaves  two 
elements  unchanged.  For  such  a  substitution  could  only  be  a  trans- 
position of  the  two  remaining  elements.  Consequently  the  roots  of 
the  corresponding  equation  of  the  fourth  degree  are  all  rational 
functions  of  any  two  among  them.  But  the  group  cannot  be  writ- 
ten in  the  form 

for  it  contains  only  substitutions  of  the  two  types  (XiX.,Xo}  and 
(a?ia?2)  (^a^^i),  so  that  the  orders  n^  and  n^  can  only  be  2  and  3,  while 
tijna  must  be  equal  to  12. 

§  196.  If  however,  the  degree  of  the  equation  of  §  194  is  a 
prime  number  p,  we  have  precisely  the  case  treated  at  the  begin- 
ning of  the  Chapter. 

To  show  this  we  observe  that  by  Theorem  II,  the  transitive  group 
G  of  the  equation  is  of  an  order  which  is  a  divisor  of  p{p  —  1). 
Since  the  transitive  group  is  of  degree  p,  its  order  is  also  a  multi- 
ple of  p.  It  contains,  therefore,  a  substitution  Sj  of  the  p^^  order, 
and  consequently  also  a  subgroup  of  the  same  order.     If  now  in 

§  128,  Theorem  I,  we  take  «  =  1,  and  put  for  r  the  order  ^-^ \ 


RATIONAL    RELATIOJtS    BETWEEN    THREE    ROOTS.  227 


it  follows  that  k  —  0,  that  is,  G  contains  no  substitutions  of  order  p 
except  the  powers  of  -s^, .  Consequently  we  must  have  s^s^sf  '  =  s^^, 
and  this  is  the  assumption  made  in  §  188. 

Equations  of  this  kind  were  first  considered  by  Galois,*  and 
have  been  called  Galois  equations.  We  do  not  however  employ  this 
designation,  in  order  to  avoid  confusion  with  the  Galois  resolvent 
•equations,  /.  c,  those  resolvent  equations  of  which  every  root  is  a 
rational  function  of  every  other  one. 

If  a  substitution  of  the  group  G  of  an  equation  of  the  pres- 
ent type  is  to  leave  any  element  x.^  unchanged,  we  must  have  from 
§193 

{k^  —  1  )z  -\-  ok^  +  ^  ^y  {i—k)  "0     (mod.  p). 

Since  k^  —  1  is  either  --0  (mod.  p)  or  is  prime  toj),  it  follows  that 
either  every  ,r  is  unchanged  and  the  substitution  is  equal  to  1,  or 
•one  element  at  the  most  remains  unchanged. 

Tlieoreiii  III.  //  all  the  roots  of  an  irreducible  equation 
of  prime  degree  p  are  rational  functions  of  two  among  them,  the 
group  of  the  equation  contains,  besides  the  identical  substitution, 
p  —  1  substitutions  of  order  p  and  substitutions  luhieh  affect  p)  —  \ 
elements.  The  solution  of  the  equation  reduces  to  that  of  two 
Abelian  equations. 

§  197.  The  simplest  example  of  the  equations  of  this  type  is 
furnished  by  the  binomial  equation  of  prime  degree  p 

in  the  case  where  the  real  p^^^  root  of  the  absolute  value  of  the  real 
•quantity  A  does  not  belong  to  the  domain  of  those  quantities  which 
we  regard  as  rational. 

The  roots  of  this  equation,  if  X^  is  one  of  them,  are 

The  quotient  of  any  two  roots  of  the  equation  is  therefore  a  power 
•of  the  primitive  p^^  root  of  unity  w.  A  properly  chosen  power  of 
this  quotient  is  equal  to  <o  itself.  Consequently  if  any  two  roots 
,r^  and  Xy  are  given,  every  other  root  x^  is  defined  by  an  equation 

*  Evariste  Galois*  Oeuvres  niath(?niatiques,  edited  by  Liouville  in  Vol.  11  of  the 
Journal  de  inatli^inatiqiies  pures  et  appliquees,  184C.   pp.  381-414. 


228  THEOEY    OF    SUBSTITUTIONS. 


\Xyj  ^y 

that  is,  Xa,  is  a  rational  function  of  x^  and  Xy. 

As  soon,  therefore,  as  it  is  shown  that  the  equation 

x^  —  A  =  0 

is  irreducible,  it  is  clear  that  it  belongs  to  the  type  under  discussion. 

If  the  polynomial  x^  —  A  were  factorable 

x^'  —  A  =  (p,{x)<poXoL^..., 

then,  since  p  is  a  prime  number,  the  several  factors  could  only  be 
of  the  same  degree,  if  they  were  all  of  degree  1.  The  roots  would 
then  all  be  rational.  Consequently  this  possibility  is  to  be  rejected. 
Suppose  then  that  <Pi{x)  is  of  higher  degree  than  crof-^)-  Let 
the  roots  of  <Pi{x)  and  (foi^)  be 

<fi [X) ;     X  I,  X  2i  00  2,  .  .  .  X  ,f^j 

^2\'^)  7         OP     ^^   OC     21   OC     r^,    .    .    .   X     „„. 

Then  the  last  coefficient  of  each  of  the  polynomials  c^j  and  -<p2 

±  X\   X'2  X'.    .  .  .   =  oj^'^  .T/'i »        /       ^       -. 
+  X"2X"2X",  ...=  oj-,x,"-^,       ^''^  ^  "^'^ 

and  consequently  their  quotient 

±oj^xr,   (w>o) 

is  rational  within  the  rational  domain.  Since  p  is  a  prime  number^ 
it  is  possible  to  find  an  integer  /j.  such  that  the  congruence 

or  the  equation 

ma  :z=  -yp-j-  1 

shall  be  satisfied.     Then  the  quantity 

(±  Xi"uJ-'Y  =  ±  iC^^  +  ^wMT—  ^  A''XiOjl^-'=  ±  A^x, 

and  consequently  x/,  is  rational.  From  the  reducibility  of  the 
equation  would  therefore  follow  the  rationality  of  a  root,  which  is 
certainly  impossible. 

The  group  of  the  equation  is  of  order  p{p  —  1).  For  if  we  leave 
one  root  Xi  unchanged,  any  other  root  ojXi  can  still  be  converted 
into  any  one  of  the  p  —  1  roots  (ox^j  oj^x^ ,  oj'^Xi  .  .  .  w^~'a?i. 


RATIONAL    RELATIONS    BETWEEN    THREE    ROOTS.  229 

Theorem  IV.     The  binomial  equation 

in  which  A  is  not  the  p^^  power  of  any  quantity  belonging  to  the 
rational  domain,  belongs  to  the  type  of  §  196.  Its  group  is  of  order 
p(p  —  l). 

§  198.  Remark.  By  Theorem  III  every  irreducible  equation 
the  roots  of  which  are  rational  functions  of  two  among  them  is  alge- 
braically solvable.  At  present  we  have  not  the  means  of  proving  the 
converse  theorem.  It  will  however  be  shown  in  the  following  Chapter 
by  algebraic  considerations,  and  again  at  a  later  period  in  the  treat- 
ment of  solvable  equations  by  the  aid  of  the  theory  of  groups,  that 
every  equation  of  prime  degree,  which  is  irreducible  a)id  algebrai- 
cally solvable,  is  either  an  equation  of  the  type  above  considered,  or 
an  Abelian  equation.  Before  we  pass  to  such  general  considera- 
tions, we  treat  first  another  special  case,  characterized  by  rational 
relations  among  the  roots  taken  three  by  three. 

§  199.  An  equation  is  said  to  be  of  triad  character,  or  it  is 
called  briefly  a  triad  equation*  if  its  roots  can  be  arranged  in  tri- 
ads Xa,Xp,  Xy  in  such  a  way  that  any  two  elements  of  a  triad  deter- 
mine the  third  element  rationally,  i.  e.,  if  Xa  and  Xp  determine  Xy, 
x^  and  Xy  determine  Xa,  and  Xy  and  Xa  determine  x^. 

Thus  the  equations  of  the  third  degree  are  triad  equations;  for 

Xi  -j-  X>  -J-  £^3  ^^=  Cj . 

Of  the  equations  of  the  higher  degree,  those  of  the  seventh  degree 
may  be  of  triad  character.      In  this  case  the  following  distribution 

of  the  roots  x^ ,  Xo,  .  .  .  Xj  is  possible : 

Xi ,  X2 )  "^3  j       X\,  X^ ,  X^  I       J?i ,  Jyg ,  Xi  \      X2 ,  X4 ,  Xf, ',       X2 ,  .^5 ,  X'j ', 
"^3^  Xif  Xf',       X^ ,  X^,  X^,  ♦ 

If  the  degree  of  an  equation  is  n,  there  are         „ —    pairs  of  roots 

Xa,Xp..     With  every  one  of   these  pairs  belongs  a  third  root  Xy. 

Every  such  triad  occurs  three  times,  according  as  we  take  for  the 

original  pair  of  roots  x^^  Xp-,  x^,Xy',  or  a:-^? '^a-      There  are  there- 

fi(fi I ) 

fore   ^ triads,  and  since  this  number  must  be  an  integer,  it 

♦Noether:  Math.  Ann.  XV,  p.  89. 


230  THEORY    OF     SUBSTITUTIONS. 

follows  that  the  triad  character  is  only  possible  when  n  =  6i)i  + 1  or 
n  =  6m  -|-  3 .  The  case  n  =  Qin  must  be  excluded,  because  n  must 
be  an  odd  number,  as  appears  at  once  if  we  combine  .^i  with  all  the 
other  elements,  which  must  then  group  themselves  in  pairs. 

The  general  question  whether  every  n  =  Qm-\-l,  n  —  Qm-^-S- 
fnTnishes  a  triad  system  we  do  not  here  consider.  It  is  however 
easy  to  establish  processes  for  deducing  from  a  triad  system  of  n 
elements  a  second  triad  system  of  2n  -j-  1  elements,  and  from  two- 
triad  systems  of  >?i  and  Wg  elements  a  third  system  of  n^  n.2  elements. 
From  the  existence  of  the  triad  character  for  7i  =  3  follows  therefore 
that  for  n  =  7,  15,  31,  .  .  . ;  9,  19,  39,  .  .  . ;  21,  43,  .  .  .  These  do  not 
however  exhaust  all  possible  cases.  There  are  for  example  triad 
systems  for  n  ■=  13,  etc. 

§  200.  We  proceed  to  develop  the  two  processes  above  men- 
tioned. In  the  first  place  suppose  a  triad  system  of  n  elements 
a\,  Xo,  .  .  .  x„  given.  To  these  we  add  n-f  1  other  elements  x^; 
x\,  ic'a,  .  .  .  x'  „.  We  retain  the  ln{7i  —  1)  triads  of  the  former  ele- 
ments, and  also  construct  from  these  — ^-^ new  triads  by  accent- 
ing in  each  case  every  two  of  the  three  .t's.  Finally  we  form  n 
further  triads  Xq,  Xi,  x\',  Xq,  x'r,,  x'2',   .  .  .  ,  and  have  then  in  all 

4.n(n—l)         _{2n  +  l)2n 

6      '^''~      6: 

triads,  which  furnish  the  system  belonging  to  the  2n  -\-  1  elements. 
For  example,  suppose  n  =  3.     We  obtain  then  the  following  sys- 
tem 

X^  y  X2  ^  X-^',        -l^j  ,  J?  2  )  "^  3  5        ^  1  )  ^2  5  '^*  3  )        ^  1  ?  ^  2  >  "^S  5        '^'0  5  •^^H  '^  1  5 
'^'(1  ?  '^2  ?  "'^  2  ?       ^0  ?  ^^3  ?  '^  3  > 

which  agrees,  apart  from  the  mere  notation,  with  the  triad  system 
for  seven  elements  established  in  the  preceding  Section. 

§  201.  Again,  suppose  two  triad  systems  of  degrees  n,  and  r/g  to 
be  given.  The  indices  of  the  first  system  we  denote  by  a ,  6 ,  c ,  .  .  . , 
those  of  the  second  by  a,  fi ,  y,  .  .  .  We  may  designate  a  triad  by 
the  corresponding  indices.  Suppose  that  the  triads  of  the  first 
system  are 


RATIONAL  RELATIONS  BETWEEN  THREE  ROOTS.  231 

^^i)  a,  b,  c;     a,d,e;     b  ,d,g;    .  .  . 

and  those  of  the  second 

We  denote  the  elements  of  the  combined  system  by  a?«a,  a?«/3,  x^a,  ..  . 
and  form  for  these  a  triad  system  as  follows.  In  the  first  place,  we 
write  after  every  index  of  T^)  the  index  a.     In  this  way  there  arise 

_i}^-L____Z  triads  of  elements  with  double  indices. 

D 

In  the  same  way  we  write  /?,  then  y,  then   ^,  .  .  .   after  every 
index  of  T^).     We  obtain  then  in  every  case  \ and  in  all 


6 


h(ni  — 1) 
6 


triads  of  the  elements  x„ 


1  '^a  /3  )  '^ay  ) 


^6 a  J  <^6^! 


All  of  these 


are  different  from  one  another.     They  are 

aa,  da^  ea\  ba^  da^  ga', 
nf:t,  dfi,  ey9;  bfi,  d^ ,  giS-, 
cf'r,    <ir,    ey,     by,  dy,  gy, 


r,) 


aa,    bo.^    ca 
ay,    by,  cy 


Again,  we  write  every   index  of   the  system   Tj)  before  every 
index  of  Tj),  and*  obtain 

^2(^2 — 1) 

triads  among  the  same  n^n^  elements  with  double  indices.      These 
are  also  different  from  one  another  and  from  those  of  T'^    They  are 

a '/ ,  a /5^ ,  ay]     aa,  ao,  as;     a « ,  a C,  ar^;  .  .  . 
ba,  bfj,    by;     6a,  5^,    bs;     ba,  b^,  b-^;  .  .  . 


T",) 


ea,    Cti,    cy,      Ca^   Ct),    GS', 


ca 


crj'. 


Finally  we  combine  every  triad  of  T^)  with  every  triad  of  T2)  by 
writing  after  the  three  indices  of  a  triad  of  T^)  the  three  indices  of 
a  triad  of  Tg).  With  any  two  given  triads  this  can  be  done  in  six 
ways.     For  example  from  b,d,  g  and  ct,  ^j^fj  we  have 

ba,d:,gry,      ba,drj,g:;      bZ,da,gr^,      b^,drj,ga;      brj,da,gC-y 

br^,  dC,  ga. 


232  THEORY    OF    SUBSTITUTIONS. 

We  obtain  therefore  from  Tj)  and  To) 


^ni(ni  — 1)^2(^2 — 1)             %W2  —  rii  —  Wg  +  1 
6 g -^ =  n,n, g 


such  combinations.     These  are  again  all  different  from  one  another 
and  from  those  of  T'g)  and  T'^).     They  are  the  following: 


r"s)  i 


aa,  6/5,  cy;  aa,  by,  ciS;  a;?,  5a,  Gy;...ay,  bfi,  c«; 
a«,  bd,  C£;  auj  be,  cS;  ad,  ba,  cs-,  .  .  ,  as,  bd,  ca; 
aa,  dl3,  ey;     aa,  dy,  e/9;     a/5,  da,  ey;  .  .  .  ay,  djS,  ea-. 


We  have  therefore  now  constructed  in  all 

^lini 1)  ^2(^2 1)    ,  71^712  —  Ui  —  ^2  + 1  _  ^,^2(^1^2  —  1) 

n, g         +  n,        J—  +  n,n2  ^  -  -^ 

different  triads  among  the  elements 

The  three  tables  Tg)  therefore  form  a  possible  triad  system  for  niU^ 
elements. 

§  202.  The  triad  group  for  n  =  8  demands  no  special  notice.  It 
is  simply  the  symmetric  group  of  the  three  elements. 

To  determine  the  group  of  the  triad  equation  for  n  =  7  we  pro- 
ceed as  follows,  restricting  ourselves  to  irreducible  equations  of  this 
type. 

With  this  restriction  the  resulting  group  of  7  elements  is  transi- 
tive. Its  order  is  therefore  divisible  by  7,  and  it  consequently  con- 
tains a  circular  substitution  of  the  7*^  order,  which  we  may  assume 
to  be 

oj  —  \0C^0u20C^i)u^3u^UC^t)u-j). 

We  determine  now  conversely  the  arrangements  of  the  7  elements  in 
triads,  which  are  not  disturbed  by  the  powers  of  s, .  These  must  be 
such  that  if  Xa,  x^,  Xy  form  a  triad,  the  same  is  true  for  every  a?a  +  **, 
Xp-\-i,Xy-\-i  (i  =  1 ,  2,  .  . .  6).  Again  with  a  proper  choice  of  nota- 
tion we  may  take  Xa  =  Xi,  Xp  =  X2,  since  a  proper  power  of  Sj  will 
contain  the  two  elements  Xa,  Xp  in  succession.  If  now  we  apply  the 
powers  of  Sj  to  the  system 

Xi,  a?2 >  "^3 5     *^i  J  '^2 >  ♦^4 ?     Xi,  X2,  x^;     Xi,  X2f  X(^',  x^,  X2 ,  x^j ; 


KATIONAL  RELATIONS  BETWEEN  THREE  ROOTS.  233 

•it  appears  that  only  tlie  second  and  the  fourth  cases  give  rise  to  a 
triad  distribution  of  the  required  character,  viz. 

Ml)     X'l,  X2,  x^;     x.2f  x^,  x^j     x^j  .^4,  Xqj     X4,  ii?5,  x-;',     X-,,  x^j  x^] 

.1*6  ,    X-J   ,     X.)  5  Xjy     Xly     X^\ 

Xf, ,  x~, ,  x^ ;     Xf ,  Xi,  Xr^ ; 

The  two  distributions  are  not  essentially  different,  each  being  ob- 
tained from  the  other  by  interchanging  x>,  d?^;  x^,  x^',  and  Xi,  Xr^. 
We  may  therefore  assume  that  Tj)  is  given,  and  that  s^  belongs 
to  the  corresponding  group.  If  there  are  other  substitutions  of  the 
7*^  order  belonging  to  the  group,  a  proper  power  of  every  one  of 
these  will  contain  ic,  and  x.^  in  succession.  We  may  write  the  sub- 
stitution therefore  "^ 

(xiX2Xa.^x„^x„^x,,^x„^)  =  (1  2  tta  cti  a^  cif,  a-,) 

To  this  substitution  correspond,  as  in  the  case  of  s^ ,  only  two 
triad  systems,  which  proceed  respectively  from  1,  2,  a^  and  1,  2  a^. 
The  indices  a^,  .  .  .  a^  must  be  so  taken  that  the  new  systems  coin- 
cide with  T,).  In  this  way  we  obtain  seven  new  substitutions  s. 
For  example,  if  the  seven  triads 

1,  2,  a,,;  2,  ag,  a^;  ag,  a^,  1;  a^,  a.,,  2;  a,-,  ae,  ag;  a^,  a^,  a^ ;  a^,  1,  a,,  " 

are  to  coincide  respectively  with 

1,2,4;     2,3,5;     3,7,1;     7,6,2;     0,4,3;     4,5,7;     5,1,6, 

we  must  have    a^  =  3,    a^  —  7 ,    a^  =  6,    a^  =  4,    a^  =  5,  and  accord- 
ingly s  =  {x^X2XQX^x^x^X:).     Similarly  we  obtain  for  the  seven  new  s's 

.S'2  =  (XiX2Xr^X^X.^-,X(i),    6*3  =  {x^X^X^X^X-jX^Xr,),    S4  =  {XiX^X-.X^X^X^X^, 

05  —  yXyXqijC^X'^XQ^x^Xi^f^  og  —  \XiX2XrpCffiC'jX ^x^j ^  s-j  ^^^^  {XiX^x^Xepc^x^x-j)^ 
og  —  yXiXoX-^x^XKpCj^x^). 
Beside  the  powers  of   .s, ,  82 ,  .  .  .  Sg  there  can  obviously  be  no  other 
substitutions  of    the    V^  order   in  the   group.       We  note,  without 
further  proof,  that  it  follows  from  this  by  the  aid  of  §  76,  Theorem 
XII,  that  the  required  group  is 

I  tS*]  ,  §2  ,   .  .  .  Sg   j- 

The  same  result  has  been  obtained  by  Kronecker  from  an  entirely 
different  point  of  view. 


234  THEORY    or    SUBSTITUTIONS. 

Theorem  V,     The  roots  of  the  most  general  in^editcible  triad 
equation  of  the  7'^  degree  can  be  arranged  as  follows: 

j's  =  '^,(iro,  Xi),  Xi  =  >'^,{x, ,  x^),  X-,  =  ^^Xo,  x^),  x^  —  //i(u'3,  x,), 

The  group  of  the  equation  is  tlie  Kronecker  group  of  order  168^. 
defined  by 

\z     az-i-b\,     \z     aO{z-\-b)  +  c\ 

(a  =  1,  2,  4;     5,  c  =  0,  1,  ...  6;     0{z)  =  —z\z'  +  1) ) 

It  is  doubly  transitir>e.      Those  of  its  substitutions  which  replace 

Xq  j  Xi  by  a?; ,  a?3  are 

yXQXiX^)  {X.iX^X^),    [XqXiX^)  {X2P0qX^),     [XqXiX^)  {^X2X^X^),    (XqXiX^)  {X^X^X^f.. 

All  these  also  replace  x^  by  Xq.     Consequently  ive  have  also 

and  similarly 

x^  =  '?i(a?2 ,  Xi),     X2  =  ^^x, ,  X,) ;  etc. 
All  the  substitutions  of  the  group  ivhich  interchange  Xq  and  x^  are 

yXQXi)  (X^Xr^)^       yX^fXl)  {X^XQ),        (^S^q^jJ  {^X2XiX^XQfj        yX^Xif  yX2XQXQX^), 

and  since  these  all  leave  x^  unchanged,  it  folloivs  that 

and  the  same  property  holds  for  all  the  other  triads.      Every  sym- 
metric function  of  the  roots  of  a  triad  is  a  1 -valued  resolvent. 

§  203.  We  examiDG  also  the  triad  equations  for  n  =  9.  In  the 
construction  of  the  triads  it  is  easily  recognized  that  there  is  only 
one  possible  system,  if  we  disregard  the  mere  numbering  of  the  ele- 
ments. We  can  therefore  assume  the  system  to  be  that  constructed 
in  §201,  and  designate  the  elements  accordingly  by  two  indice8> 
each 


00,  10,  20 
00,01,02 
00, 11,  22 
00, 12,  21 

A  characteristic  property  of  every  such  triad 

pq,  p'q\  p"q" 
is  the  condition 


01,11,21;  02,12,22; 

10,11,12;  20,21,22; 

01,12,20;  02,  10,21: 

01,10,22;  02,11,20: 


RATIONAL  RELATIONS  BETWEEN  THREE  ROOTS.  235 

B)  p  -h p'  -^p"^*!  +  q'  +  q" ^- 0     (mod.8) 

From  this  it  follows  that  every  substitution 

s  -—p,q     ap  -i-  bq  +  a ,  a'p  +  h'q  -j-  a  \ 

transforms  the  triad  system  into  itself.  For  the  indices  p,  g;  p\q'r 
p",  q"  become  respectively 

ap  +  bq  -{-  a ,  a'p  -\-  b'q  +  a' ; 
ap'  -f  hq'  -\-  a ,  a'p'  +  '^'q'  +  *'^'  \ 
ap"  +  bq"  +  a,     ay  +  b'q"  +  «": 

and  if  the  condition  B)  is  satisfied  by  p,p',p"',  q,  q',  q'\  it  is  also 
satisfied  by  the  new  indices. 

Conversely,  every  substitution  that  leaves  the  triad  system  un- 
changed can  be  written  in  the  form  .<?  by  a  proper  choice  of  the 
coefiicients  a,  b,  a;  a,  6',  a'.  For  if  f,  is  any  substitution  of  the 
triad  group  which  replaces  the  index  (0,  0)  by  ('/,  a'),  then 

S,  =\p,q       p  +  ^^^   g  +  «'  I 

does  the  same,  and  consequently  ^2  =  ^1^1  \  which  also  belongs  ta 
the  group,  leaves  (0,0)  unchanged.  If  now  to  replaces  (0,  1)  by 
(  6,  b' ),  then 

s,  =   p,q     ap  +  bq,  a'p  +  b'q  \ , 

where  a  and  a  are  arbitrary,  will  leave  (0,  0)  unchanged,  and  will 
replace  (0,  1)  by  b,  b'.  Consequently  /..  =  U^.,  ^  will  leave  both  (0,  0) 
and  (0,  1)  unchanged.     Again  if  ^3  replaces  (1,  0)  by  (c,  c'),  then 

^3  =  I P,  q      cp,c'p-i-q\ 

will  leave  (0,  0)  and  (0, 1)  unchanged,  and  will  replace  (1,  0)  by  (c,c') 
consequently   t^  =  tyS^^~\  which  belongs  to  the  group  of   the  triad 
equation,  will  leave  (0,  0),  (0,  1)  and  (1,  0)  unchanged.     A  glance  at 
the  triad  system  shows  that  we  must  have  /*  =  1 ,  and  it  follows  accord 
ingly  that 

Consequently  t^  is  actually  of  the  assigned  form.  Remembering 
further  that  we  have  established  in  §  145  the  necessary  and  sufiicient 
condition  that  this  form  shall  actually  furnish  a  substitution,  we  have- 
the  following 


:236  THEOllY    OF    SUBSTITUTIONS. 

Theorem  VI.  The  group  G  of  the  irreducible  triad  equa- 
tion of  degree  9,  consists  of  all  the  substitntions 

s  :^^  i  2),  q       cip  ~\-  bq  -j-  « ,  ap  -\-  b'q  +  o.'  i     (mod.  8) 
ab'  —  a'b^O     (mod.  3) 

The  order  of  (J  is,  from  §  145 

r=r3^(3"    -1)(3^  — 8)==27.16 

The  roots  of  the  equation   are  connected,  in  accordance  with  the 
triad  system,  as  follows  : 

'^•20^^^       ('^30  5  '■^10)5     '^'21  "~  'M'^'oi?  '^1]})     •'^li  —  'm.^02  5   ■^'12)'')  •    •    • 

All  the  substitutions  of  G  which  replace  0*00  and  x^q  by  a^n,  and  x^^ 
^?'e  of  the  form 

s'  1:3  j  2J,  q       p  +  6(/  +  1,  b'q  \     (mod.  3), 

and  since  these  all  convert  x..^  into  Xoo,  it  foUoivs  that  we  have  also 

All  the  substitutions  of  the  group  which  interchange  .^oo  and  x^q  are 
of  the  form 

s"  —  j  27,  q        2p  -f  bq  +  1,  b'q  \     (mod.  3), 

-and  since  these  all  leave  .r^o  unchanged,  ive  have,  again, 

•5*20  ^^  '^(,"'"00 »  "'^lo)  —  '^('^'lO?  •^0*1)5      •    •   • 

§  204.  The  arrangement  in  triads  given  at  the  beginning  of  the 
preceding  Section  possesses  a  peculiarity,  which  we  can  turn  to 
:account.  The  triad  system  is  so  distributed  in  four  lines  that  the 
three  triads  of  every  line  contain  all  the  9  elements. 

Evidently  every  substitution  of  the  group  permutes  the  several 
lines  as  entities  among  themselves.  We  determine  now  those  sub- 
stitutions which  convert  every  line  into  itself.     If 

'^—  Pi  ^1  <^P  +  ^^1 4-  "■ ,  ct'p  -\-  ^'(1  -\-  "■'  I  (mod.  3) 
is  to  convert  the  lirst  line  into  itself,  the  new  value  of  q  must  depend 
:solely  on  the  old  value  of  q,  but  not  on  the  value  of  p.  Consequently 
we  must  have  a  =  0.  If  the  substitution  is  to  convert  the  second 
line  also  into  itself,  we  must  again  for  the  same  reason  have  b  =  0. 
The  substitution  is  therefore  of  the  form 

fT  —  \p,q       ap  -h  « ,   b'q  -f  0/       mod.  ?>). 


RATIONAL    RELATIONS    BETWEEN    THREE    ROOTS.  237 

That  conversely  all  these  substitutions  satisfy  these  two  conditions- 
is  obvious.     Their  number  is  3" -2^,  since  ah' .'lO  (mod.  3). 

It  is  further  required  that  t  shall  also  leave  the  third  and  fourth 
lines  unchanged.  The  third  line  has  the  property  that  in  every  triad 
(pq,  p'q',  xj"q")  the  three  sums 

p-\-q,  p'-\-q\  p"-\-q" 
have  respectively  the  values  0,  1,  2  (mod.  3);  the  fourth  that 

P  +  ^-  P'+g':-p"+^"^    (mod.  3). 
If  now  we  apply  <t  to  the  triad  (00, 12,  21)  of  the  fourth  line,  we 
obtain 

( a  ,  o: ;      a  +  a ,    26'+  o! ;      2rt  -f  a  ,    //  +  a')      (mod.  3), 

and  consequently  we  must  have 

./  +  a'  ^  -  a  +  26'  4-  a  +  a'  z:  2a  -f  }>'  +  o.  +  a'     (mod.  3), 

that  is, 

a      b'     (mod.  3). 

The  linal  form  of  (j  is  therefore 

n=    p^q     ap-\-a^   aq-\-a   \. 

Conversely  all  the  substitutions  of  this  type  convert  every  one  of 
the  four  lines  into  itself.  The  substitutions  form  a  subgroup  H  of 
G  of  order  2  •  3",  since  a  can  only  take  the  values  1  and  2.  if  is  a 
self -conjugate  subgroup  of  G.  For  if  r  is  any  substitution  of  (r, 
then  r  ^  i/r  leaves  every  line  unchanged,  /.  e.,   r^^  Hr  =z  H. 

The  group  H  of  order  2  •  3',  being  a  self- conjugate  subgroup  of 
G  which  is  of  order  2*  •  3^,  we  can  construct,  by  §86,  the  quotient 
T=  (t\H  of  order  2^-  4  =  24  and  of  degree  4  (corresponding  to 
the  four  lines  of  triads).  This  group  is  of  course  the  symmetric 
group  of  4  elements. 

If  therefore  we  construct  a  function  y?  of  the  9  elements  x, 
which  belongs  to  the  group  ii,  this  four-valued  function  is  the 
root  of  a  general  equation  of  the  fourth  degree,  the  group  of  which 
is  T. 

If  this  equation  of  the  fourth  degree  has  been  solved,  the  group 
of  the  triad  equation  reduces  to  H^  of  order  2  •  3^,  as  is  readily 
apparent.  The  systematic  discussion  of  this  class  of  questions  is 
however  reserved  for  Chapter  XIV. 


238  THEORY    OF    SUBSTITaTIONS. 

§  205.  We  consider  further  the  subgroup  of  H  which  leaves 
every  single  triad  of  the  first  line  of  our  table  unchanged.  In 
order  that 

\p,q      ap-i-a,   aq-{-a'  \ 

may  have  this  property,   the  values    q  =  0,  1,  2,  must  give  again 
g  =  0,  1,2.     Consequently  «e^0,  a  =  1,  and  we  must  take 
r=   p,  q        pJ^a,    q\. 

The  r's  form  again  a  self-conjugate  subgroup  of  I  of  H  of  order  3. 
We  construct  by  j^  ^^"^  the  quotient  U  —  H:  I.  U  is  of  order  3  •  2 
and  of  degree  3,  corresponding  to  the  three  triads.  U  is  therefore 
the  symmetric  group  of  three  elements.  If,  then,  we  construct  a 
function  c''  of  the  9  elements  ,r,  which  belongs  to  the  group  /,  this 
(after  adjunction  of  cr)  three-valued  function  depends  on  a  general 
equation  of  the  third  degree. 

If  the  latter  has  been  solved,  the  group  of  the  triad  equation 
reduces  to  /.     Accordingly  the  symmetric  functions  of 

are  known,  and  therefore  those  three  values  depend  on  an  equation 
of  the  third  degree,  the  coefficients  of  which  are  rationally  express- 
ible In  terms  of  <!' .  This  equation  is,  in  fact,  an  Abelian  equation, 
since  its  group  only  permutes  the  roots  cyclicall3^     We  have  then 

Theorem  VFI.  The  irreducible  triad  equations  of  degree 
9  can  be  solved  algebraically. 

§  206.     In  close  relation  to  the  above  stands  the  following 

Theorem  VIII.  //  three  of  the  roots  of  an  irreducible 
equation  of  the  9"'  degree  are  connected  by  the  equations 

,i:,  ^  0(,r^ ,  .r.)  =  (>(x, ,  .r,),  ■    .i\  =  0(,x', ,  x,)  =  0(x, ,  ,r,,), 
.r,=  0{x^,.r,)  =  0{x,,x,), 

in  which  0  is  a  rational  function  of  its  two  elements,  the?)  the  c(/ua- 
tion  can  be  solved  algebraically. 

We  consider  the  group  of  the  equation.  It  is  transitive,  and  it 
replaces  the  three  roots,  x^^ ,  x_ ,  x.  by  three  others  between  which 
the  same  relation  must  exist  as  between  a?i,,r2,.r.j  themselves. 
Suppose   the  new   roots   to  be  x\,  x'.,,  x'...     If    the   two   systems 


RATIONAL    RELATIONS    BETWEEN    THREE    ROOTS.  239 

,Ti ,  ,r. ,  .r;i ,  and  x\,x'2,  x'^,  have  two  roots  in  common,  then  they 
have  also  the  third  root  in  common.  For,  if  Xi  =  x\ ,  .r^  =  x\ ,  it 
follows  that 

^nd  if  x'..  and  Xj  are  not  the  same  root,  the  given  equation,  having 
«qual  roots,  would  be  reducible. 

If  x^  is  a  root  different  from  a',,  x.,,  .r.,,  there  is  a  substitution  in 
the  group  which  replaces  x\  by  .r^.  If  this  substitution  leaves  no 
element  unchanged,  we  obtain  an  entirely  new  system  x^ ,  x-, ,  ^^. 
But  if  one  element,  for  example  x., ,  remains  unchanged,  we  have  for 
anew  system  x.,,  .r^,  x^.  Proceeding  in  this  way,  and  examining  the 
possible  effects  of  the  substitutions,  it  is  seen  that  all  the  roots 
arrange  themselves  in  the  triad  system  of  9  elements.  Comparing 
this  result  with  Theorem  VI,  it  appears  that  the  equation  is 
exactly  one  of  the  triad  equations  just  treated. 

It  is  known  *  that  the  nine  points  of  inflection  of  a  plane  curve 
of  the  third  order  lie  by  threes  on  straight  lines.  These  lines  are 
twelve  in  number,  and  four  of  them  pass  through  every  point  of 
inflection.  Any  two  of  the  nine  points  determine  a  third  one,  so  that 
the  points  form  a  triad  system,  as  considered  above.  The  abscissas 
or  the  ordinates  of  the  nine  points  therefore  satisfy  a  triad  equa- 
tion of  the  9th  degree,  and  this  equation,  belonging  to  the  type 
above  discussed,  is  algebraically  solvable. 

It  can,  in  fact,  be  shown  that  if  ^'i,x,,  X:~  are  the  abscissas  or 
the  ordinates  of  three  points  of  inflection  lying  on  the  same 
straight  line,  then 

X;  =  0{xi,X2),  x,  =  0{x2,x.^,  x,  =  0(x.„Xi), 

where  0  is  a  rational  and  symmetric  function  of  its  two  elements. 
The  discussion  of  this  matter  belongs  however  to  other  mathemati- 
cal theories  and  must  be  omitted  here. 

*0.  Hesse:  Crelle  XXVIII,  p.  G8;  XXXIV,  p.  101.    Salmon:  Crelle  XXXIX,  p.  365, 


CHAPTER  XIII. 


THE  ALGEBRAIC  SOLUTION  OF  EQUATIONS. 

§  207.  In  the  last  three  Chapters  various  equations  have  been- 
treated  for  which  certain  relations  among  the  roots  were  a  priori 
specified,  and  which  in  consequence  admitted  the  application  of  the 
theory  of  substitutions. 

In  general  questions  of  this  character,  however,  a  doubt  presents 
itself  which,  as  we  have  already  pointed  out,  must  be  disposed  of 
first  of  all,  if  the  application  of  the  theory  of  substitutions  to  gen- 
eral algebraic  questions  is  to  be  admissible.  The  theory  of  substi- 
tutions deals  exclusively  with  rational  functions  of  the  roots  of 
equations.  If  therefore  in  the  algebraic  solution  of  algebraic  equa- 
tions irrational  functions  of  the  roots  occur,  we  enter  upon  a  re- 
gion in  which  even  the  idea  of  a  substitution  fails.  The  funda- 
mental question  thus  raised  can  of  course  only  be  settled  by  alge- 
braic means;  the  application  to  it  of  the  theory  of  substitutions, 
would  beg  the  question.  To  cite  a  single  special  example,  proof 
of  the  impossibility  of  an  algebraic  solution  of  general  equations 
above  the  fourth  degree  can  never  be  obtained  from  the  theory  of 
substitutions  alone. 

§  208.  In  the  discussion  of  algebraic  questions  it  is  essential 
first  of  all  to  define  the  territory  the  quantities  lying  within  which 
are  to  be  regarded  as  rational. 

We  adopt  the  definition*  that  all  rational  functions  with  integral 
coefficients  of  certain  quantities  9i',  ^J{",  91'",  .  .  .  constitute  the- 
rational  domain  (9f{,  9f{"  3??'",  .  .  . ).  If  among  any  functions  of  this 
domain  the  operations  of  addition,  subtraction,  multiplication,  divis- 
ion, and  involution  to  an  integral  power  are  performed,  the  result- 
ing quantities  still  belong  to  the  same  rational  domain. 

The  extraction  of  roots  on  the  other  hand  will  in  general  lead 

*Ii.  Kronecker:  Herl.  Ber.  1879,  p.  205  If.;  cf.  also:  arithm.  Theorie  d.  algeb.  Grossen. . 


THE    ALGEBEAIC    SOLUTION    OF    EQUATIONS.  241 

to  quantities  which  lie  outside  the  rational  domain.  We  may  limit 
ourselves  to  the  extraction  of  roots  of  prime  order,  since  an  {mnY^ 
root  can  be  replaced  by  an  m^^  root  of  an  n^''  root. 

All  those  functions  of  9i',  91",  9^'",  .  .  .  which  can  be  obtained 
from  the  rational  functions  of  the  domain  by  the  extraction  of  a 
sinojle  root  or  of  any  finite  number  of  roots  are  desifrnated,  collect- 
ively, as  the  algebraic  f unctions  of  the  domahi  (91',  91",  91"',  .  .  .). 
In  proceeding  from  the  rational  to  the  algebraic  functions  of  the 
domain,  the  first  step  therefore  consists  in  extracting  a  root  of  prime 
order  p„of  a  rational,  integral  or  fractional  function  i^^v(91',  9t",  91"' .  .  . ) 
which  in  the  domain  (^R',  9t'',  91'",  .  .  .)  is  not  a  peifect  p„*^  power. 
Suppose  the  quantity  thus  obtained  to  be  V^  so  that 

yf''  =  i'V(9^',9l",9t"',..,). 

We  will  now  extend  the  rational  domain  by  adding  or  adjoining  to 
it  the  quantity  Vy,  so  that  we  have  from  now  on  for  the  rational 
domain  (Vv\  91 ,  9t",  9t"',  .  .  .),  i.  e.,  all  rational,  integral  or  fractional 
functions  of  F^,  9^',  91",  9i"',  .  .  .  are  regarded  as  rational.  The 
present  domain  includes  the  previous  one.  With  this  extension  goes 
a  like  extension  of  the  property  of  reducibility.  Thus  the  function 
xP  — Fv{^\  9t",  .  .  .)  was  originally  irreducible:  it  has  now  become 
reducible  and  has,  in  the  extended  domain  (F„-,  9i',  91",  .  .  .),  the  ra- 
tional factor  X —  Vy. 

The  new  domain  can  be  extended  again  by  the  extraction  of  a 
second  root  of  prime  order.  We  construct  any  rational  function 
which  is  not  a  perfect  {py_^,^'^  power  within  (Fv;9i',  91",  . .  .,  and 
denote  its  (p^_i)^*^  root  by  Fv_i,  so  that 

It  is  not  essential  here  that  Vy  should  occur  in  -P„_,  If  now 
we  adjoin  Vy_^,  we  obtain  the  further  extended  rational  domain 
(Fj,_i,  Vy\  91',  91",  .  .  .).     Similarly  we  construct 

V''L-'=t\_,(V,_„  n_„  n;  SR',  3!",  .  . .), 


16 


242  THEORY    OF     SUBSTITUTIONS. 

where  the  F's  denote  rational  functions  of  the  quantities  in  paren- 
theses, and  Pi,P2)  •  •  ■  Pv-2  are  prime  numbers. 

Any  given  algebraic  expression  can  therefore  be  represented  in 
conformity  with  the  preceding  scheme,  by  treating  it  in  the  same 
way  in  which  the  calculation  of  such  an  expression  involving  only 
numerical  quantities  is  accomplished. 

§  209.  The  Fa's  are  readily  reduced  to  a  form  in  which  they  are 
integral  in  the  corresponding  F's,  that  is  Va^i,  Va+2  •  •  •  ^v?  and 
are  fractional  only  in  the  9ft',  91",  .  .  . 

Thus,  suppose  that 

where  Gq,  Gi,  Gg,  .  .  . ;  H^.H^.H^,  ...  are  rational  in  F^ +2?  Fa+3,... 
Vv\  91',  91",  ...  If  now  i<^  is  a  primitive  (pa  +  i)'^root  of  unity,  the 
product 

i'a  +  l-^ 

\  =  o 

is  a  rational  function  of  Va+2,  Va+s^  .  •  •  V^;  91',  91", .  .  .  For  on  the 
one  hand  the  product  is  rational  in  the  if  s,  and  on  the  other  it  is 
integral  and  symmetric  in  the  roots  of 

n;v =^«+i(^a+2, . . .  T^v;  9t',  91", . . .) 

and  is  therefore  rational  in  the  coefficient  Fa 4-1  of  this  equation. 

Again,  if  we  omit  from  the  product  P)  the  factor  Hq-\-Hi  Va  +  i 
4-H2  F\+i  +  . . . ,  the  resulting  product 

Pa+l  —  ^ 

is  integral  in  V^+i  and  rational  in  'Fa+2?  •  •  •  ^v'^  9i'?  9i"»  •  •  •  More- 
over, since  (o  does  not  occur  in  P)  or  in  the  omitted  factor,  it  does 
not  occur  in  P,).  » 

If  now  we  multiply  numerator  and  denominator  of  Fa  by  Pj), 
the  resulting  denominator  is  a  rational  function  of  Va+2»  •  .  •  Vpi 
91',  9^", . . .   alone,  while  the  numerator  is  rational  in  these  quan- 


THE    ALaEBRAIC    SOLUTION   OF   EQUATIONS.  243 

titles  and  in  Va+i-  Dividing  the  several  terms  of  the  numerator 
by  the  denominator,  we  have  for  the  reduced  form  of  Fa 

where  the  coeffiaients  Jq)  «^n  <^2»  •  •  •  are  all  rational  functions  of 
Va+2i ' '  '    ^v\  Sfi',  31",  ...     On  account  of  the  equations 

we  may  assume  that  the  reduced  form  of  F^,  contains  no  higher 
power  of  Fa  +  i  than  the  (pa+i —  l)^^ 

The  several  coefficients  J  can  now  be  reduced  in  the  same  way  as 
Fa  above.  By  multiplying  numerator  and  denominator  of  their 
fractional  forms  by  proper  factors,  all  the  J's  can  be  converted  into 
integral  functions  of  Fa  +  o  of  a  degree  not  exceeding  Pa+2  —  1?  ^^^ 
with  coefficients  which  are  rational  in  Ta+sj  •  .  •  yv\  «R'j  3fl",  ...  In 
this  way  we  can  continue  to  the  end. 

§  210.  We  have  now  at  the  outset  to  establish  a  preliminary 
theorem  which  will  be  of  repeated  application  in  the  investigation  of 
the  algebraic  form  peculiar  to  the  roots  of  solvable  equations.* 

Theorem  I.  J/  /,,/,,..  ./^.i;  i^  are  functions  within  a 
definite  rational  domain,  the  simultaneous  existence  of  the  two  equa- 
tions 

B)  iv^—F  =0, 

requires  either  that  one  of  the  roots  of  B)  belongs  to  the  same  rational 

domain  with  /o,/i,  .  .  .  fjj-i'-,  F^  or  that 

If  all  thp/o,/j,  .  .  ./^j_]  are  not  equal  to  0,  the  equations  A)  and 

B)  have  at  least  one  root  w  in  common.  In  the  greatest  common 
divisor  of  the  polynomials  A)  and  B)  the  coefficient  of  the  highest 
power  of  w  is  unity,  from  the  form  of  B).  Suppose  the  greatest 
common  divisor  to  be 

C)  (Pq  -f  (f^w  4-  c^w"^  -jr  .  .  .-Jrwr 

Equated  to  0,  this  furnishes  v  roots  of  B).     If  one  of  these  is  deno- 

*Tliis  tlieorein  \v  is  orij^iaaily  ijiveii  by  AiujI:  ()c;iivie-i  coiiiiJiCies  II,  l<jij.  Ivroii- 
ecker  was  ibe  first  to  establish  it  iu  tbe  lull  imporiauce:  Berl.  lier.  1879,  p.  20U, 


244  THEORY   OF   SUBSTITUTIONS. 

ted  by  u\,  and  a  primitive  p^^  root  of  unity  by  cu,  then  all  the  ^ 
roots  of  C)  can  be  expressed  by 

Apart  from  its  algebraic  sign,  c^o  is  the  product  of  these  roots 

Now  since  ^  is  a  prime  number,  it  is  possible  to  find  two  numbers 

u  and  V,  for  which 

pw+v?;=  1, 
and  consequently 

One  root,  o/hL\ ,  of  the  equation  B)  therefore  belongs  to  the  given 
rational  domain. 

§  211.     We  apply  Theorem  I  first  to  the  further  reduction  of 

If  Jk  is  anyone  of  the  coefficients  «/,,  Jo^  •  •  •  >  which  doos  not  vanish. 
we  determine  a  new  quantity  Wa  +  i  by  the  equation 

annex  to  this  the  equation  of  definition  for  Va^-i 
and  fix  for  the  rational  domain 

It  follows  then,  if  A)  and  B)  of  Theorem  I  are  replaced  by  A^)  and 
Bj),  that,  since  the  possibility  W^j^^  —  0,  J^  =  0  is  excluded,  we  must 
have 

where  «>  is  a  (pa^-i)*^  loot  of  unity. 

We  can  therefore  introduce  into  the  expression  for  Fa  in  the 
place  of  Fa+i  the  function  T^a +  1,  provided  we  adjoin  the  (/>a+i)"'root 
of  unity,  ">,  to  the  rational  domain.  From  A,)  and  C^)  it  is  clear  that 
(TFa  +  r,  V«+2,...;  9l'JR",...)and(F«+,,F«+„  ...;  9^'.  91", .  .  . ), 
define  the  same  rational  domain,  and  the  equation 


.      THE   ALGEBRAIC    SOLUTION    OF    EQUATIONS.  245 

can  be  taken  in  the  scheme  of  §  208  in  place  of  Bj).  The  equations 
of  definition  for  Va^  Va-u  -  -  ■  V-i  are  not  essentially  affected  by  this 
change.  We  have  only  to  substitute  in  the  func.ions  Fa-,  i^a-n  •  •  •  -^i 
in  the  place  of  Va+i  the  value  taken  from  Cj).  The  expression  for 
Fa  then  becomes  simplified 

We  may  ^ippose  this  reduction  to  have  been  effected  in  the  case  of 
every  Fa. 

§  212.     We  pass  now  to  the  investigation  of  the  form  of  the  roots 
of  algebraically  solvable  equations.     Criven  an  algebraic  equation 
1)  fix)  =  0 

of  degree  n,  the  requirement  that  this  shall  be  algebraically  solv- 
able can  be  stated  in  the  following  terms: — from  the  ratioLal  domain 
(9^',  ^If",...),  which  includes  at  least  the  coefficients  of  1),  we  are  to 
arrive  at  the  roots  of  1)  by  a  finite  number  of  algebraic  operations, 
viz.  addition,  subtraction,  n^ultiplication,  division,  raising  to  powers, 
and  extraction  of  roots  of  prime  orders.  One  of  the  roots  of  1 ) 
can  therefore  be  exhibited  by  the  following  scheme: 

v:r,'=F..,(v.',^\w\...) 
2)       <!  v:^Lr=F^-2{v^-^^,y^',^',^"'-') 


where  G^,  G^  G^,  .  .  .  are  integral  functions  of  V^,  V^,  .  .  .  F^and  ra- 
tional functions  of  Di',  3fl",  .  .  . ,  and  G,  may  be  assumed  to  be  1(§  211). 
Taking  the  powers  of  Xq  and  reducing  in  every  case  those  pow- 
ers of  Ti  above. the  {p^  —  iy^\  we  obtain  for  every  •> 

If  these  powers  of  x^  are  substituted  in  1),  we  have 

A)        f{xo)  =  Ho-{-H,v,-^H,v;'+ . ...  +H^,-iy^i-s 

where  the  jff's  are  formed  additivily  from  the  G^""s  and  the  coeffi- 
cients of  1).     Joining  with  A)  the  equation  of  definition  of  Vi 


246  THEORY    OF    StJBSTITTJTiONS. 

and  applying  Theorem  I  to  A)  and  B),  we  have  only  two  possibili- 
ties:  either  a  root  of  B)  is  rational  in  the  domain  (F2,  V^,  .  .  .  V  ; 
iR',  9{",  .  . . ),  or 

Both  cases  actually  occur.  In  the  former  the  scheme  j),  by  which 
we  passed  from  the  original  rational  domain  to  the  root  Xq,  can 
be  simplitied  by  merely  suppressing  the  equation 

'    and  adding  the  Pi*^  root  of  unity  to  the  rational  domain. 

§  213.     As  an  example  of  this  case  we  may  take  the  equation  of 
the  third  degree 

f{x)  =  x'  —  dax  —  2b  =  0, 

the  rational  domain  being  formed  from  the  coefficients  a  and  b.  "By 
Cardan's  formi.ila 


This  algebraic  expression  can  be  arranged  schematically  as  follows: 


V^^  =  b''  —  Q; 


V^:=^b^-V,^ 

V,'  =  b—V,, 

The  expression  for  J/(a:o),  formed  as  in  the  preceding  Section,  then 
becomes 

y(x,)^--aV,  +  (F/-a)F,  +  V,V,'  =  0. 

Comparing  this  with 

and  determining  Vx  from  the  last  two  equations,  we  obtain 

^'     "      a^+(6+F3)F2  — aF/ 

so  that  Fi  is  already  contained  in  the  rational  domain  (F2,  F3;  a,  b). 
If  we  now  transform  Fj  into  an  integral  function  of  Fj  by  the  pro- 
ceis  of  §  20y,  we  obtain  from  the  relations 


THE    ALGEBRAIC    SOLUTION    OF    EQUATIONS.  247 

[a'+  (6  +  V,)<oV,—aio'V,']  [a'+  (6  +  V,)<o'V,—  aajV,''] 

=  26(6+^3)  (a +T^A 

[a(6+  V,)-d^V,-^{b-V,)V,'][2b{b-j-  V,){a+  V,')-] 

=  4.ab\b-{-V,)V,\ 

where  "*  is  a  primitive  cube  root  of  unity,  the  simpler  form 


_4ab'{b-i-V,)V,'_   a 


46^(6+ Fa/  6+l/V 

Removing   F3  from  the  denominator  by  multiplying  both  terms  of 
the  fraction  by  b —  V^,  we  have  finally 

CI/ 

and  herewith  the  reduced  form  of  Xq 

a 
Fi  can  therefore  be  suppressed  in  the  scheme  above. 

§  214.     We  return  now  to  the  results  of  §  212  and  examine  the 
second  possible  case.     In 

f{xo)  =  H,  +  H,V,-{-H,V,'+  . .  .  +iy^,_:F,^i-^ 

suppose  that  Fj  is  not  rational  in  the  domain  ( F2 ,  Fn  •  •  • ;  9^'j  9t", . . .). 
Then  from  Theorem  I 

If  now,  in  analogy  to  3),  we  form  the  expressions 

3')      x,=  G„+  G,«',*r,  +  G,WVl  +...  G^_,»"''-'''y^'-', 

(fc  =  0,l,...p,-]) 
in  which  Wj  is  a  primitive  p,*^  root  of  unity,  it  follows,  with  the  same 
notation  and  process  as  in  §  212,  that 


X 


=  G^  +  G^^'n„'-V,  -f •  Gi^'aj^'V,'  +  .  . . , 


Since  the  JHT's  vanish  identically,  the  latter  expression  is  also  equal 
to  0,  i.  e.,  Xu  is  a  root  of  f{x)  =  0  for  A;  =  1,  2,  . . .  Pi  —  I. 

For  example,  in  the  case  of  the  equations  of  the  third  degree 


248  THEORT   OF   SUBSTITUTIONS. 

x^  —  ^ax—2b  =  0, 

where  the  first  of  the  two  possibilities  above  has  been  excluded  by 
reducing  Xq  to  the  form 


the  other  two  roots  are 


CL 

h V. 


V  = 2 ) 


a 

§  215.  If  now  we  make  the  allowable  assumption  (§  211)  that 
Gi=  \,  (whereupon  Vi  may  possibly  take  a  new  form  difPerent  from 
its  original  one),  we  obtain  by  linear  combination  of  the  pi  equa- 
tions for  XqjXj,  .  .  .  x^_x 

X,  =  Oo-}-ojV,  +  G,a.,'V;'+  . . .  +  G^,_,wr^-'V,^^-\ 

x^_,  ^  G,-i-ojP^~'V,-^G.y(^^- ')¥,'-{-  .  .  .+G^^_M''-'^'V,^'-\ 
the  value  of  F, : 


The  irrational  function  V^  of  the  coefficients  is  therefore  a  rational 
and,  in  fact,  a  linear  function  of  the  roots  Xq^Xi,  .  .  .  x^^^-i  as  soon 
as  the  primitive  Pi"^  root  of  unity  «>i  is  adjoined  to  the  rational 
domain. 

§  210.  In  the  construction  of  the  scheme  2)  it  is  not  intended 
to  assert  that  Fa  necessarily  contains  Fa_i,  Fa_.2,  •  •  .  If  Va-i  is 
missing  in  Fa,  another  arrangement  of  2)  is  possible;  we  can  replace 
the  order 

V:XV=Fa  +  dVa  +  ,,...),     V'^  =  i^a(F«  +  „...),     Ff^^  =  F.  (F., .  . .) 

by  the  order 

v:-=F.^,{Va+„ . . .),  n»+r  (^«+2'  •  •  •).  vi'-'=F^_,(v^, . . .). 

It  is  therefore  possible,  for  example,  that  different  F's  occur  at  the 
end  of  the  series  2).     In  this  case  different  constructions  3)  for  the 


THE    ALGEBKAIC    SOLUTION    OF    EQUATIONS.  24© 

root  Xq  are  possible,  aiid  the  theorem  proved  in  the  preceding  Sec- 
tion holds  for  the  last  V  of  2)  in  every  case. 

To  prove  the  same  theorem  for  all  F's  which  occur,  not  in  the 
last,  but  in  the  next  to  the  last  place  in  2),  we  will  simply  assume 
that  i^i  actually  contains  V2.  The  proof  (§  215)  of  the  theorem  for 
Vi  was  based  on  the  fact  that  an  expression 

satisfied  an  equation  with  rational  coefficients.  We  demonstrate  the 
same  property  for  an  expression 

If  we  suppose  all  the  permutations  of  the  roots  of  the  equation 
1)  to  be  performed  on 

"-[jiS---]': 

fc  =  0 

the  product  of  the  resulting  expressions  is  an  integral  function  of 
y,  with  coefficients  which  are  symmetric  in  the  x^s  and  are  therefore 
rational  functions  of  91',  91",  .  .  . 

If  we  denote  this  function  by  <p{y),  the  coefficients  of  the 
equation 

<p{y)  =  0 

belong  to  the  domain  (W,  91",  .  •  .),  and  one  of  the  roots,  with  possi- 
bly an  unessential  modification  of  the  meaning  of  V2  (cf.  §  211)  is 


2/0 


=  [^2  "^1""'^  J' =  "^^^  =  F,  (F,,  .  .  . )  =  A  +  F2  +  AF/  +  . 


It  is  therefore  essential  that  V^  should  actually  occur  in  F^ .  We 
can  now  apply  to  c(y)  =  0  with  the  root  2/0  the  same  process  which 
we  applied  above  to  f(x)  =  0  with  the  root  Xq.  If  we  assume,  as  is 
allowable,  that  the  series  F^,  Fy_i,  .  .  .  F3,  V2  is  so  chosen  that  V^ 
is  not  rational  in  the  preceding  F's,  it  follows  that 

fc=:0 

where  m^  is  a  primitive  p^^^  root  of  unity.      Every  y/^  is  produced 
16a 


250  THEORY   OF     SUBSTITUTIONS. 

from  2/0  by  certain  substitutions  among  the  x^,  x^,  .  .  .  x,,_^;  conse- 
quently V2  is  a  rational  integral  function  of  the  Pi^^  degree  of  the 
roots  of  f{x)  =  0,  provided  the  quantities  Wj  and  w,  a^^e  adjoined 
to  the  rational  domain. 

In  the  same  way  every  V  can  be  treated  which  occurs  in  the  next 
to  the  last  but  not  in  the  last  place  in  2).  Proceeding  upward  in 
the  series  we  have  finally. 

Theorem  II.  The  explicit  algebraic  function  Xq  ,  ivhich  sat- 
isfies a  solvable  equation  f(x)  =  0,  can  be  expressed  as  a  rational 
integral  function  of  quantities 

V„  V,,  v„ . . .  v., 

with  coefficients  ivhich  are  rational  functions  of  the  quantities 
9t',  91".  The  quantities  V\  are  on  the  on:e  hand  ratio7ial  integral 
functions  of  the  roots  of  the  equation  f(x)  =  0  and  of  primitive 
roots  of  unity,  and  on  the  other  hand  they  are  determined  by  a 
series  of  equations 

V/«  =  i/XFa_,,  F«_2,  .  .  .  F,;  3t',  91",  .  .  . ). 

In  these  equations  the  pi,p2,Piy  .  . . py  are  prime  numbers,  and 
Fi,F2,  .  .  .  Fy  are  rational  integral  functions  of  their  elements  V 
and  rational  functions  of  the  quantities  9i',  91",  .  .  .,  ivhich  determ- 
ine the  rational  domain. 

§  217.  This  theorem  ensures  the  possibility  of  the  application 
of  the  theory  of  substitutions  to  investigation  of  the  solution  of 
equations.  It  furnishes  further  the  proof  of  the  fundamental  prop- 
osition : 

Theorem  III.  The  general  equations  of  degree  higher  than 
the  fourth  are  not  algebraically  solvable. 

For  if  the  n  quantities  x^^x^,  .  .  .x„,  which  in  the  case  of  the 
general  equation  are  independent  of  one  another,  could  be  algebra- 
ically expressed  in  terms  of  9t',  9i",  .  .  . ,  then  the  first  introduced 
irrational  function  of  the  coefficients,  Vy,  would  be  the  p^^  root  of 
a  rational  function  of  9^',  91",  .  .  .  Since,  from  Theorem  II,  Vy  is  a 
rational  function  of  the  roots,  it  appears  that  Vy,  as  a  p^- valued  func- 
tion of  Xi,  £^25  •  •  •  ^«?  thep^"^  power  of  which  is  symmetric,  is  either 
the  square  root  of  the  discriminant,  or  differs  from  the  latter  only  by 


THE    ALGEBRAIC    SOLUTION    OF    EQUATIONS.  251 

a  symmetric  factor.  Consequently  we  must  have  p^  =  2  (§  56). 
If  we  adjoin  the  function  F^  =  /Sj  V  ^  to  the  rational  domain,  the 
latter  then  includes  all  the  one- valued  and  two-valued  functions  of 
the  roots.  If  we  are  to  proceed  further  with  the  solution,  as  is  nec- 
essary if  n  >  2,  there  must  be  a  rational  function  Vy_yof  the  roots, 
which  is  (2j)t,_i)- valued,  and  of  which  the(p^_i)^^  power  is  two-val- 
ued. But  such  a  function  does  not  exist  if  n  >  4  (§  58).  Conse- 
quently the  process,  which  should  have  led  to  the  roots,  cannot  be 
continued  further.  The  general  equation  of  a  degree  above  the 
fourth  therefore  cannot  be  algebraically  solved. 

§  218.     We  return  now  to  the  form  of  the  roots  of  solvable  alge- 
braic equations 

3)  x,=  G,-\-  F,4-  <^.Fr-h  .  .  .  -\-G,^_,V,^^^-\ 

We  adjoin  to  the  rational  domain  the  primitive  Pi"',  p.^-^,  .  .  .  roots  of 
unity,  and  assume  that  the  scheme  which  leads  to  a\,  is  reduced  as 
far  as  possible,  so  that  for  instance  Va  is  not  already  contained  in 
the  rational  domain  (Fa_,  .  .  .  V^^  9i',  9i",  .  .  . ;  Wj,  <'a,,  .  .  .  ).  We 
have  seen  that  the  substitution  of 

o>,^V,     {k  =  l,2,...p,-l) 

for  V  in  8)  produces  again  a  root  of  f{x)  =  0.  We  proceed  to  prove 
the  generalized  theorem: 

Theorem  IV.  //  in  the  scheme  2),  ii'hich  leads  to  the  expres- 
sion 3)  for  a^o,  any  V^  is  multiplied  by  any  root  of  unity,  the  values 
Fa-M  ^^a- J?  •  •  •  1^2)  ^1  ^^**^^  *^*  general  be  converted  into  new  quan- 
tities Va,  Va-i,  •  •  •  ^'21^1-  ^/  ^'^^  latter  are  substituted  in  the  place 
of  the  former  in  the  expression  for  x^,  the  result  is  again  a  root 
offix)  =  0. 

We  may,  without  loss  of  generalit}^,  assume  that  f{x)  is  irredu- 
cible in  the  domain  {W ,  3t'',  .  .  .  ). 

Starting  now  from  3),  and  denoting  by  cUr  ,a  primitive  r*^^  root 
of  unity,  we  construct 

JJ{x-x,)  =  JJlx-iGo-\-<V,+  G,<o,''V;^+  .  .  .)]. 

\-0  A.=.0 

In  this  product  V^  certainly  vanishes.     Possibly  other  F's  vanish 


252  THEOEY    OF    SUBSTITUTIONS. 

also.  Suppose  that  Fa(a>^2)  is  the  lowest  V  that  actually  occurs. 
Then 

4)       JJ  (^  —  ^\)='fa{00',Va,    Fa  +  i,..   ••)  =  «0  +  «1  "^a  +  «2  "^a' +    •   -. 
A  =  0 

The  a's  which  occur  here  belong  to  the  domain  (Fa  +  n  •  •  •)•  ^® 
construct  further 

ra  - 1  ' 

\  =  0 

where  Vj,  {b^a-{-l)  is  again  the  lowest  V  that  actually  occurs. 
Similarly  let 

Pb-l 
A.  =  0 

and  finally,  supposing  the  series  to  end  at  this  point, 

Pc  —  1 
A  =  0 

where /d  is  rational  in  91',  W,  .  .  .  ,  all  the  F^+i)  -  •  .V^,  disappearing 
with  V,. 

We  assume  now,  reserving  the  proof  for  the  moment,  that  the 
functions 

Mx;  n,  . .  .),  Mx',  y„  .  .  .),  fXx;  K.,  .  .  .),  Ux',  3^',  .  .  .) 

are  irreducible  in  the  domains 

(T-«,n+i . . .),  (n,  n+i . .  •),  (K.,  y.+,, . . .),  (r,  r'. . .), 

respectively.  Then  fa{x;  91',  .  .  .)  =  0  and /(a?)  =  0  have  in  the  do- 
main (91',  9t",  .  .  .)  one  root  x  =  Xq  in  common,  since  x — Xq  occurs 
as  a  common  factor  of  fd{x)  and  f(x).  Both  these  functions  being 
by  assumption  irreducible,  it  follows  that 

'/,(a:;i»',r',...)=/H- 
If   now  we  assign  to    V^  any  arbitrary   value  tv  consistent  with 
Vl"  =^ F^,{'3i' , .  .  .  )    and   to    V^_i    any  value   tV-i   consistent   with 
V^'/S^^  =  Fv_^{Vy',  91', .  ,  •),  and  continue  in  this  way,  we  have  thej 
series 


2') 


THE    ALGEBKAIC    SOLUTION    OF    EQUATIONS.  253 


3')  -- 0  =  9i>  +  Q'l  ^1+  92Vi^  +  .  .  •  +  (/in  - 1^''' '  '^ 

3')  being  obtained  from  3)  by  putting  the  r's  in  place  of  the  F's. 

The  product 

A  =  <» 

will  only  differ  from  those  obtained  above  by  the  introduction  of  the 
gr's  and?y8  in  place  of  the  6r's  and  F's,  since  in  all  the  reductions  2') 
replaces  2).  Consequently  this  product  is  equal  to/„(^';  r,,,  r„+i, .  .  .) 
and  similarly 

A=0 
Pb—\ 

A.=  0      , 
Pc—l 

JJMx',  coM; ^.+,, . . .)  =  .«^;  '^i". ^)^"',  •  •  •) -^/(^•)- 

A  =  0 

This  furnishes  the  proof  that  r^  is  a  root  of  f{x)  =  0. 

We  have  still  to  prove  the  irreducibility  of  fa{x),  fh{x),  ...  in 
the  rational  domains  (F,,,  F„  +  i,  .  .  .)?  (^/o  Vh  +  i,  •  •  •)•  •  •  ?  respect- 
ively. 

Assuming  the  irreducibility  of  f„{x)  in  the  domain  ( V,, ,  y„  + 1 , . . .), 
we  proceed  to  demonstrate  that  of  fo{x)  in  the  domain  (Vi,  F^,^  j , . . .). 
The  method  employed  applies  in  general. 

If  (f{x',  Vb,  .  .  .)  is  one  of  the  irreducible  factors  of  fi{x),  so 
chosen  that  it  contains  fa{x',  F„,  .  .  .)  as  a  factor,  then  we  have  in 
the  domain  V„ ,  F^  + 1 ,  .  .  .  the  equation 

8)  <p{x',  y„ . . .)  =/(x;  y«, . . .) . </ix;  y„, . . .), 

which  can  be  rewritten  in  the  form 


254  THEORY    or    SUBSTITUTIONS. 

A)  ^o  +  7i  V^  +x,V:-^  .  .  .  +7,^_,  F/«-\ 

where  the  /'s  are  rational  functions  of   V„^^,  .  .  .     Since  moreover 

B)  T^^-i^'«(F„  +  ,,...)-0, 

it  follows  from  Theorem  I  that  either  F„  is  rational  in  F„^i,  .  .  . , 

which  is  to  be  excluded,  or  all  the  /'s  vanish,  so  that  A)  and  with  it 

the  equation  8)  above  still  hold,  if  F„  is  replaced  by  w„  V„ ,  <'>„"'  F,, ,  .  .  . 

Again,  f„{x;  iof^V„,  .  .  . )  is  different  from  fu{x;  ojf^V,,,  .  .  .).    For  if  we 

write 

faix;  F„,  .  .  .)  -  s,-i-,,V,,  +  .,y/+  .  .  . , 

it  would  follow  from  the  equality  of  the  two  functions  /„  that 
and  consequently,  from  the  equation  of  definition 

that  Va  must  be  rational  in  the  domain  (F„  + 1,  .  .  .  ^}t',  .  .  .  (Wj,  .  .  .), 
since  "^py^. 

Accordingly /„(ir;  y„,  .  .  .  j,  fa{x;  w„F„,  ...),..  .  are  all  divisors  of 
(f.  All  these  functions  are  different  from  one  another,  and  they  are 
all  irreducible  in  the  domain  (F„,  F^,  ^j,  .  .  .  ).  Consequent!}^  e  con- 
tains their  product,  which,  on  account  of  the  degrees  of  /„  and  fi  in 
A',  is  possible  only  if  (f  and/^  coincide. 

Since  the  foregoing  proof  holds  for  every  irreducible  factor  of 
1),  it  still  holds  if  we  drop  the  assumption  of  irreducibility. 

§  219.  At  the  beginning  of  the  preceding  Section  we  remarked 
that  in  the  product  construction  with  F,  other  T"'s  might  vanish. 
This  possibility  is  however  excluded  in  the  case  of  certain  T's,  as 
we  shall  now  show. 

We  designate  any  Vr  of  2)  as  an  external  radical  when  the  fol- 
lowing Ft  +  i?  Ft 4  2,  .  •  .  i-  (i-,  Fr-^i,F^  +  :,  .  .  .  do  uot  contain  Vr- 
Every  such  external  radical  can  be  brought  to  the  last  position  of 
2),  and  the  expression  of  .r,,,  as  given  in  3),  can  be  arranged  in 
terms  of  every  external  radical  present.  We  shall  see  that  in  the 
product  construction  with  V^  no  other  external  radical  can  be 
missing.  Thus,  if  F/  is  missing  in 
?',  - 1 


THE    ALGEBEAIC    SOLUTION    OF    EQUATIONS.  255 

then  /„  cannot  be  changed  if  we  replace  Vr  in  the  fundamental 
radical  expression  by  oj^'^Vj,  without  thereby  changing  V^.  If,  as 
a  result,  the  (xs  are  converted  into  the  g's,  we  should  then  have  also 

A  =  0 

Every  linear  factor  in  x  of  this  last  expression  must  therefore  be 
equal  to  some  factor  of  the  preceding  expression 

A)  S'o+  V,-\-g,V{+  ...=  Go-i-<o^'V,  +  t?,^,^nV-f  .  •  . 

Taking  into  account  the  equation  of  definition 

it  follows  from  Theorem  I  that  either  V^  is  rational  in  Fo,  F3,  .  .  .  «>t, 
which  may  be  excluded,  since  otherwise  2)  could  be  reduced  fur- 
ther on  the  adjunction  of  o^,  or  that 

In  some  one  of  these  equations  Vr  must  actually  occur.  Develop- 
ing this  equation  according  to  powers  of  F^,  we  have 

A,)    K,-i-K,Vr-{-K,Vj'-{-  .  .  .  =:  K,^K,co,Vr  +  K,co,'Vr'-h  .  .  .  , 
and  combining  with  this 

B,)  V/r  —  Fr(Vr+,,...)=0, 

the  impossibility  of  both  alternatives  of  Theorem  I  appears  at  once. 
Consequently  Vr  could  not  have  been  missing  in  the  product  con- 
struction. 

If  we  consider  only  /„  (§218,  4)),  the  series  2)  ending  with  V„ 
can  also  contain  external  radicals,  in  fact  possibly  such  as  are  not 
external  in  respect  to  the  entire  series.  These  also  cannot  vanish 
in  the  further  product  construction.  The  irreducibility  of  /«  being 
borne  in  mind,  the  proof  is  exactly  the  same  as  the  preceding. 

Theorem  V.  In  the  pi-oduct  construction  of  the  preceding 
Section  no  external  radicaU  can  disappear  from  /„  except  V^..  The 
same  is  true  for  fi,  in  respect  to  the  external  radicals  occurring 
among  F^,  F^+u  •  •  •  V^,  and  so  on. 

If  several  external  radicals  occur  in  x^  or  in  one  of  the  ex- 
pressions f  a,  fh  ,fci  '  '  •  y  tf^  product  of  all  the  corresponding  expo- 
nents is  a  factor  of  n. 


256 


THEOKY    OF    SUBSTITUTIONS. 


Theorem  VI.  //  an  irreducible  equation  of  prime  degree 
p  is  algebraically  solvable,  the  solution  icill  contain  only  one  exter- 
nal radical.  The  index  of  the  latter  i^  equal  to  p,  and  if  lo  is  a 
primitive  p"'  root  of  unity,  the  polynomial  of  the  equation  is 

V    -  1 


Theorem  VII.     //  the  algebraic  expression 

is  a  root  of  an  equation  f(x)  —  0,  ivhich  is  irreducible  in  the 
domain  (9t',  9t",  .  .  .  )>  <^^^  ^f  ^^  construct  the  product  of  the  p^ 
factors,  in  which  Fj  is  replaced  by  oj^V^,  ^o^V^  .  .  . 


f,{x;  Y,,,  .  .    )=jl  {^  —  Xj,), 


where  V„ is  the  lowest  V present,  and  again  the  product  fb(x;  Vj,,...) 
of  the  p„  factors  f„(x;  o)„^V,,,  .  .  .  ),  and  so  on,  ive  come  finally  to 
the  equation  f{x)  —  0,  the  degree  of  which  is 

n  =  p^p„Pb .  ■  • 
The  functions  f„  ,fi,...  ar^e  irreducible  in  the  domains  {V„,V„^i,.. .), 

§  220.  We  examine  now  further  those  radicals  which  vanish  in 
the  first  product  construction.  The  remaining  V^,  F«  ^  j ,  .  .  .  are 
not  altered  in  the  product  construction.  We  may  therefore  add 
these  to  the  rational  domain,  or,  in  other  words,  we  may  consider  an 
irreducible  equation /(a?)  =/„(x;  F,,,  ..  .)   in  the   rational   domain 

(F„,y„  +  „...;3{',r',...)- 

Here  all  the  V^,  Vo,  .  .  .  V„__^  already  vanish  in  the  first  product 
construction. 

We  examine  now  what  is  the  result  of  assigning  to  F„_i  any 
arbitrary  value  consistent  with  its  equation  of  definition,  then  with 
this  basis  assigning  any  arbitrary  value  of  V„_2  consistent  with  its 
equation  of  definition,  and  so  on.     Suppose  that  the  functions 

Va-ly   Va-2i  •   •   '    V2i   Vl'l   ^0,  G^2  J   •   -  •    ^Tp-l 

are  thereupon  converted  into 


THE  ALGEBRAIC  SOLUTION  OF  EQUATIONS.  257 

The  new  value  assumed  by  Xq  is  then 

^0  =  90+ g^^i  +  g^vi'  +  Qzv;'  +  . . .  +  gp-iv,^"'. 

From  §  218  Cq  is  again  a  root,  and  this  together  with  the  system 
$1,  ^2-  •  •  •  ^i)-i)  which  arises  from  Cq  when  v^  is  replaced  by  w^i, 
<o\,  .  . .  ojP-^v^,  gives  again  the  complex  of  all  the  roots.  We  can 
therefore  take 

go  +  ^i     +92V^'     +...=Go+   ^'^1+    G,co''V,'+.,., 

^o  +  ">'^i  +  g2«>V+  . . .  =  G^o  +  «>'"^i  +  G^2«>'"'T^i'-f  .  .  .  , 

where  w',  w",  a»'"  ,  .  .  .  are  the  p^^  roots  of  unity  w,  «>^  w^,  . .  . ,  apart 
from  their  order.     By  addition  of  these  equations  we  obtain 

9o=  Go, 
so  that  Gq  is  unaffected  by  the  modifications  of   l^«_i,  T^a-2>  •  •  •  ^2- 
Also  p  Gq  is  the  sum  of  all  the  roots,  and  is  therefore  a  rational 
function  in  the  domain  (91^,  9t",  .  .  . ). 

Again  we  obtain  from  the  system  above  the  equation 

+  G2F/K  +  ^'"^-'+^'"'">-'+ .  .  .) 
+  ... 

Here  the  first  term  on  the  right  vanishes.  We  denote  the  paren- 
theses in  the  following  terms  briefly  by  p^i,p^2i  P^si  •  •  •  >  ^^^  write 

9)  v,=P.,V,  +  P.,G,V,'  +  P^,G,v;'+... 

On  raising  this  to  the  p^^  power 

A)  v,^  =  F,{v„v„  .  .  .  v„_,',U\  ...)  =  [P^V,-j-P,G,V,'-i-  .  .  .7 

and  annexing  the  equation  of  definition 

B)  V,^  =  F,{V„  V„...  F„_,;  91',  .  .  .), 

it  follows  from  Theorem  I  that  either   Fj  is  rational  in 

^2,  F3,...K_,;  v„v,,...v„_,;  91',  9t",..., 
or  that 

Vf  =  A,,     A,  =  0,     A,  =  0,  ...A^_,  =  0. 
17 


258  THEOBY    OF    SUBSTITUTIONS. 

We  consider  now  the  first  of  these  alternatives.  In  the  rational 
expression  of  V^  in  terms  of  T^2  >  ^3  ?  ♦  •  • ;  ^2?  ^3 »  •  •  • ;  9^'  Ut" ,  .  .  .  all  the 
^2,^3,.  .  .Va-i  cannot  vanish;  otherwise  V2  should  have  been  sup- 
pressed in  2).  If  then  we  define  Fj ,  as  in  §§  208  and  212  by  a 
system  of  successive  radicals,  some  V^  will  occur  last  among  the  v^s 
and  some  V\  last  among  the  v^s.  If  we  substitute  the  expression 
for  Vi  in  Xq,  we  have 

x,=R{V„..  .  n.i;^,.  .  .)  =  R,(V„  .  .  .  V,_,',v,,  .  .  .  v„_,;^',  .  .  .) 

Here  all  the  v's  cannot  vanish,  as  we  have  just  seen.  For  the  same 
reason  all  the  T' s  cannot  vanish,  since  we  might  have  started  out 
from  ?o-  S^t  V^  and  V\  are  two  external  radicals,  and  the  product 
of  their  exponents  must  therefore  be  a  factor  of  p  (Theorem  V). 
This  being  impossible,  the  first  alternative  is  excluded. 
Accordingly  we  must  have  in  A) 

The  question  now  arises  what  the  form  of  9)  must  be  in  order  that 
its  p*^  power  may  take  the  form  Vf  =  Aq  .     The  equation  A)  is 

The  result  just  obtained  shows  that  the  left  member  is  unchanged 
if  Vi  is  replaced  by  wFi,  oj'^V^^,  .  .  .      Consequently 


and  on  the  extraction  of  the  p^^  root  we  have 

But  from  9)  follows  also 

P.OJ'^V,  +  P2G2  oj'^Vi'  +  .  . .  =  w'^^i, 
and  equating  the  two  left  members  and   applying  Theorem  I  as 
usual,  we  have 

i2,=:0,     P,G2  =  0,  .  .  .  ^K_iG^._i  =  0,     P,  +  ,G,  +  ,  =  0,  .  . . , 
that  is,  9)  reduces  to  the  single  term 
9')  V,  =  PM.V,\ 


THE  ALGEBRAIC  SOLUTION  OF  EQUATIONS.  259 

Substituting  this  result,  together  with  Qq  =  Gq  in  the  expression  for 
^oj  we  have 

On  the  other  hand  the  root  ^o,  which  is  contained  among  Xq,  Xi,  .  .  ., 
oan  also  be  expressed  in  the  form 

^0=  Go  +  <o'V,-}-G,<o''V,'-}-  G,co''v;'-i-  .  .  . , 

and,  comparing  the  two  right  members,  it  follows  from  Theorem  I 

that  terms  with  equal  or  congruent  exponents  (mod.  p)  are  identical. 

In  particular  we  have 

P^.G.V,'^=G,<o'<V,% 
and  therefore 

10)  v,  =  o/<G,V,\ 

Theorem  VIII,  If,  in  the  explicit  expression  3)  of  the  root 
iCo  of  an  irreducible  equatioyi  of  prime  degree  p,  the  irrationalities 
V  are  modified  in  any  way  consistent  with  the  equations  of  defini- 
tion, then  Fj-P  is  converted  at  the  same  time  into  (G^Vi^y^. 

§  221.  The  relations  arising  from  such  a  transformation  are 
most  readily  discussed  by  the  introduction  of  a  primitive  congru- 
ence root  e  in  the  place  of  the  prime  number  p.     We  write 

V,  =  \/b;,     G,V{:=^R„    -G,,vf^i^R„..., 

where  however  every  e°-  is  to  be  reduced  to  its  least  not  negative 
remainder  (mod.  p — 1).     Then  the  quantities 

■^1,     G,v:\     G,V,\  ...    G,_(Vr' 
coincide,  apart  from  their  order,  with  the  quantities 


and  we  have 


11)  a:o=G^o+Vi^o+Vi^i+-..  +  Vi^^-2. 

The  changes  in  the  values  of  the  radicals,  considered  above,  which 
replace  V^  by  (G^F/)^  and  consequently  [GaFi"]  by  \Go.^x^^, 
where  o.^  <  p  and 

a^^Xa     (mod.  p), 
have  therefore  the  effect  of  replacing  every  R^  by 


260 


THEORY    OF    SUBSTITUTIONS. 


i^a  +  .        (a=:0,l,   ...i>  — 2), 

where  /As  <  p —  1  and  is  defined  by  the  congruence 

e^^A     (mod.  p). 
Consequently  the  quantities 

1)  JXq,    ill,     -^2>   •  •  •  -t^p~.2 

are  converted  in  order  into 

-L^Ki     -lCk  +  1^     -rtK  +  2>    •   •   •  ■^K+p  —  21 

and,  if  the  same  operation  is  performed  a  times,  I)  is  replaced  by 

-t^aK^  -f^aK-[-lj   •^a/c-i-2)   •   •   •  -\--'^aK-\-p  —  2') 

where  the  indices  are  of  course  to  be  reduced  (mod.  p  —  1). 

If  there  is  another  modification  of  the  radicals,  which  converts 
Rq  into  Rfj,,  this  on  being  repeated  /9  times  converts  the  series  I)  into 

Finally  if  we  apply  the  first  operation  «  times  and  the  second  /5 
times,  I)  becomes 

Here  a  and  /?  can  be  so  chosen  that  ax  -\-  iS/j.  gives  the  greatest 
common  divisor  of  /-  and  /j-.  Consequently  if  i^//is  the  R  of  lowest 
index  which  is  obtainable  from  R^j  by  alteration  of  the  radicals,  every 
other  R  obtainable  from  Rq  in  this  way  will  have  for  its  index  a 
multiple  of  /?,  so  that  the  permutations  of  the  i^'s  take  place  only 
within  the  systems 

Ro,  Rk,  R9i-  •  '  ■  R(^p-i    i\j. 

Here  k  is  sl  divisor  of  p  —  1. 

There  are  then  alterations  in  the  meaning  of  the  radicals  which 
produce  the  substitution 

(-Ro  ^k  Rih  •  •  • )  (^1  Rk  -f  1  R2k  +  1  ...)... 

§  222.     The  preceding  developments  enable  us  to  determine  the 
group  of  the  irreducible  solvable  equations  1)  of  prime  degree  p. 
Every  permutation  of  the  x's  can  only  be  produced  by  the  alter- 


THE    ALGEBEAIC    SOLUTION    OF    EQUATIONS.  261 

atiohs  in  the  radicals  V^,  V2,  .  .  .  Va-x,  and  consequently  only  such 
permutations  of  the  a?'s  can  occur  in  the  group  as  are  produced  by 
alterations  of  the  y's.  From  the  result  of  the  preceding  Section 
Vi  can  be  converted  into  W'G^V^''  ,  and  the  possible  alterations  in 
Vi  do  not  change  this  form.  Substituting  this  in  the  table  of  §  215, 
we  have 


We  examine  now  whether  any  root  cc^  can  remain  unchanged  in 
this  transformation.     In  that  case  we  must  have 

Go4-  .  .  .  +^'^^G^/T^f  +  . .  .  =  G^o+  <+''G,^Vf+  .  . .  , 

and  from  the  method  which  we  have  repeatedly  employed  it  follows, 
as  a  necessary  and  sufficient  condition,  that 

/^e  ^fy--\-T     (mod.  p). 

If  e^^l  (mod.  p),  then  for  r4=0  there  is  no  solution  /x,  and 
therefore  no  root  cc^  which  remains  unchanged.  But  for  r  =  0, 
every  /j.  satisfies  the  condition,  and  the  substitution  reduces  to  iden- 
tity. 

If  e^dsl,  then  for  every  r  there  is  a  single  solution  //,  and  the 
corresponding  substitution  leaves  only  one  element  unchanged. 

Theorem  IX.  The  group  of  a  solvable  irreducible  equation 
of  prime  degree  is  the  metacyclical  group  (§  134)  or  one  of  its  sub- 
groups. . 

§  223.     Since  now,  as  we  saw  in  §  221,  all  the  substitutions  of  the 

group  permute  the  values i?o,-Rt,-R2fc?- ••  ^^^J  among  themselves,  the 

symmetric  functions  of  these  values  are  known,  and  the  values  them- 

p  — 1 
selves  are  the  roots  of  an  equation  of  degree  — —-  .     The  latter  is  an 

rC 

Abelian  equation  since  the  group  permutes  the  values  Bq,  Rk,R2k,  •  •  • 
only  cyclically.  Consequently  eyery  Kk,  Bok,  ^sk,  •  •  •  is  a  rationa 
function  of  i?o-  33ut  the  same  is  true  of  every  B^.  For  the  form 
of  the  substitution  at  the  end  of  §  221  shows  that  after  the  adjunc- 


262  THEORY    OF    SUBSTITUTIONS. 

tion  of  Ro  all  the  other  BaS  are  known,  since  the  group  reduces 
to  1. 

Finally  it  appears  that  if 

Ra  =  ^a(-Ro) 

then 

Ba  +  km  ^^  Fa\Bkm)- 

f 

For  the  application  of  properly  chosen  substitutions  of  the  group 
converts  the  first  equation  into  the  second. 

We  consider  now  all  the  substitutions  of  the  group  of  f(x)  =  0 
which  leave  Rq  =  F/  unchanged  and  accordingly  can  only  convert 
Vi  into  some  w^Fj.  Then  Xq  is  replaced  by  x^.  But  since  R^  is  a 
rational  function  of  Rq,  it  appears  that  R^  =  GJ'V^  is  also  unchanged, 
so  that  GeVi"  is  converted  into  some  (r^fy'^Vi*.  The  power  w^^  can 
be  determined  from  Xi,-,  for  the  expression  for  Xj,  contains  the  term 
GeVi^j^''%  and  this  must  be  identical  with  G^oji^Vi^  Consequently 
/x  =  ve,  and  GeV^^  becomes 

G,co-^V,% 

while  at  the  same  time   V^"  becomes 

so  that  the  factor  Ge  remains  unchanged.  That  is,  every  substitu- 
tion of  the  group,  which  leaves  Rq  unchanged,  leaves  Ge  unchanged 
also.  Accordingly  Ge  is  a  rational  function  of  Rq.  The  same  is 
true  of  all  the  other  (r's.     We  can  therefore  write 

12)  x,=  G,  +  V,+  ^,{Vf)-  V,'-^n(V,-).  v;'+...+  <p,-,{v/)-  v,^^-\ 

where  <f2i  9zi  -  -  -  are  rational  functions  of  Vf  in  the  domain 
(3fl',  Dt",  .  .  . ).  From  this  it  appears  that  in  11)  the  radicals 
\/ Rq,  \/R^,.  .  .  do  not  admit  of  multiplying  every  term  by  an 
arbitrary  root  of  unity,  as  indeed  is  already  evident  d  priori  since 
otherwise  Xq  would  have  not  p,  but  p^  values^ 

A  still  further  transformation  of  12)  is  possible.     We  have 

^r;=  Ge  y/R^'  =  c/',{Ro) .  \/R^\ 
From  §  221  there  are  alterations  in  the  ^1,^2?  •  •  •  T^a-i  which  convert 
Rq  into  R;,  and  consequently   \/ Rq  into  co'^'^/y/Rk-      The  form    of 
the  exponent  of  (o  evidently  involves  no  limitation.     At  the  same 
time  the  Xq  becomes 


THE  ALGEBRAIC  SOLUTION  OF  EQUATIONS.  263 

and  since  i?i  becomes  Rk^i,  it  follows  that  />/ Ri  becomes 


If  now  we  apply  these  transformations  to  the  equations  above, 
we  obtain    . 


We  can  therefore  also  write 

x,^G,+  ^R,  +\/^  +/C/^,+  ... 


+  ^2 W  •  ^Rf  +  v''2 W  •  ^Rf  +  02(i^2.)  •  \^i^2/  + 


Theorem  X.  The  roots  of  a  solvable  equation  of  prime 
degree  p  can  be  tvritten  in  either  of  the  two  forms  12)  or  13).     In 

13)  wRk,  ^R2k,  •  •  .  «^e  rational  functions  of  \/Ro.     The  values 

Rq,  Rj,,  R2k,  .  .  .  R(pzlL-iY 

k  I 

are  roots  of  a  simplest  Abelian  equation,  the  group  of  which  is  com- 
posed of  the  powers  of 

a  —  (Rq  Rf,  R^j, .  .  .) 

Its  roots  are  connected  by  the  relations 

14)  ^S;=/(i?o)•^/57,   ^Rr,=f{R,)-s/Rf, 
\/R;,=f{R,,)^</E^,... 

where  Gek—f{R^. 

§  224.  The  form  13),  together  with  the  relations  14)  between 
i^o?  ^/ij  •  .  .,  is  not  only  necessary  but  also  sufficient  for  a  root  Xq  of 
an  irreducible  solvable  equation  of  prime  degree  p.  For  14)  shows 
that  all  the  possible  permutations  among  the  i^'s  are  simply  powers 
oi  ff.  If  now  0-*  converts  i?o  into  Raki  then  ^  Rq  will  become 
some  o)I^\^Rq,  and  from  14)  ^  , 

^i4=/(i^o)  •  \^R7^  becomes  w^^^'fiR,) .  s/R^,  etc. 


264  THEORY    OF     SUBSTITUTIONS. 

Consequently  13)  becomes 

4-... 
That  is,  Xq  has  only  the  p  values  Xq^x^,  . .  .x^_-^. 

§  225.     We  will  examine  further  the  Abelian  equation  of  degree 

k         "^  <P{R)  =  0, 

which  is  satisfied  by  i?o,  -Ra;?  •  •  •  -R(m_i>-    From  14)  we  have 

H.. =/^(-Ra)  -/""{R,)  -r^iR,)  ■  R/\ 


and  since  R^j,  =  Rqj  it  follows  from  these  equations  that 

i=-Bo-""'-[r"'"'w-/-""'-^(i?.).../(-B,-.-.)]^ 

1  =  R.-'-'-'lf'-'-'iB,)  -f '-'-'" (R,,) . .  ./(i?„)] . 

Now  the  primitive  congruence  root  e  for  p  can  be  so  chosen  that 
€^~^  —  1  is  divisible  by  no  power  of  p  higher  than  the  first.     For  if 

e^-'=l     (mod.p') 
then 

(p  —  e)^-'=e^-''  —  (p  —  l)pe^-''^e''-'-{-peP-^=l-\-pe^-^      * 

(mod.  p'), 
80  that  (p  —  e)^~Ms  divisible  only  by  p,  and  we  can  therefore  take 
p  —  e  in   place  of  e.     In  e^~^  —  l=p.q,  then,  q  is   prime  to  p. 
Consequently  we  can  determine  t  so  that 

tq-\-1^0     (mod.  p). 
Suppose  that 

tq  =  sp  —  1. 

Substituting  this  in  equation  15),  and  taking  the  p*^  root  of  the  t^^ 
power,  we  have 

1  =  R,-"-' If -'■-'-'■  (i?„)  ./'^-'-*(iJ,) ...]', 


THE    ALGEBKAIC    SOLUTION    OF    EQUATIONS.  265 

Again,  if.  we  write 

16)  f{R^,)=aa 

and  take  again  the  p*^  root  we  have 


Since  by  equations  16)  the  tta's  are  rational  functions  of  the  roots  of 
an  Abelian  equation,  the  aja  are  themselves  roots  of  an  Abelian 
equation.     The  substitution 

(T=  {EQRj,R2k  .  .  .) 
of  the  former  corresponds  to  the  substitution 

r  =  (tto  tti  tta  .  .  . ) 
of  the  latter.     If  the  roots  aQ^a^a^,  .  .  .   are   different   from   one 
another,  then  Bak  is  a  function  of  aa 

and  this  function  is,  in  fact,  the  same  for  all  values  of  a  (§  189). 

Theorem  XI.     The  quantities  ^  R  can  he  reduced  to  the 
form 

^R,=  <P{a„)  .  [a^-'-'  ■  «/-■-''  .  . .  a„_,y, 
17)  ii/Ii,  =  *(«,)  .  la^-'-"  ■  a,''-'-''  . .  .  a,y, 


This  form  contains  the  roots  a^,  a^,  .  .  .  a,„_i  of  a  simplest  Abelian 
■equation.  'P  is  an  arbitrary  function.  The  form  17)  is  not  only 
necessary  but  also  sufficient. 

The  last  statement  remains  to  be  proved. 

In  the  first  place  the  -K's,  as  rational  functions  of  the  (distinct) 
roots  of  an  Abelian  equation,  are  themselves  roots  of  an  Abelian 
equation  w^ith  the  group  1,  <t,  ^r^,  .  .  .  <t*"~^,  which  corresponds  to  the 
group  1,  r,  T^,  ,  .  .  t"*~\     Again  the  first  two  equations  of  17)  give 


_r<p{a,)-a-n-      , 


fW-B;\ 


(p{i 

so  that  we  are  brought  back  to  the  characteristic  equations  14).* 

*Cf.  Abel:  Oeuvres,  II  pp.  217  ff.  (Edition  of  Sylow  and  Lie);  and  Kroneeker:  Mo- 
natsber.  d.  Berl.  Akad.,  1853,  June  20. 


CHAPTER  XIV. 


THE  GROUP  or  AN  ALGEBRAIC  EQUATION. 

§  226.  We  have  already  seen  in  Chapter  IX,  §  153  that  every 
special,  or  afPect,  equation  f{x)=0  is  completely  characterized  by 
a  single  relation  between  its  coefficients  or  between  its  roots.  Sup- 
pose that  in  any  particular  case  the  relation  is 

^(d?i,  a?2,  .  .  .  x^)  =  0. 

More  accurately  speaking,  it  is  not  the  function  <p  itself,  but  the 
family  of  <p  and  the  corresponding  group  G,  which  characterize  the 
equation.  Only  those  substitutions  among  the  roots  are  permissi- 
ble, which  belong  to  G.  For  this  group  we  have  the  fundamental 
theorem : 

Theorem  I.  Given  an  equation  f{x)—0  and  a  correspond- 
ing rational  domain  SI,  all  rational  integral  functions  of  the  roots 
of  the  equation  which  are  rational  within  91  are  unchanged  by  the 
group  G  of  the  equation,  i.  e.,  they  belo7ig  to  the  family  of  G  or  to 
an  included  family.  Conversely,  all  integral  functions  of  the  roots 
which  are  unchanged  by  G  are  rational  within  91. 

The  algebraic  character  of  a  given  equation,  for  example  one 
with  numerical  coefficients,  is  therefore  by  no  means  determined  by 
the  knowledge  of  the  cofficients  alone;  but,  as  was  first  indicated  by 
Abel,  and  then  systematically  elaborated  by  Kronecker,  the  bounda- 
ries of  the  rational  domain  must  also  be  designated.  The  solution 
of  the  equation  a?^  — 2  =  0,  for  example,  requires  very  different 
means,  according  as  V  2  is  or  is  not  included  in  the  rational  domain. 
The  rational  domain  can  be  defined  on  the  one  hand  by  assigning 
the  elements  91',  91",  .  .  . ,  from  which  it  is  constructed.  Or  we  may 
construct  the  Galois  resolvent  equation  and  determine  one  of  its  irre- 
ducible factors  in  the  rational  domain.  The  latter  does  not,  to  be 
sure,  entirely  replace  the  assignment  of  91',  91",  .  . . ,  but  it  furnisher 


THE  GKOUP  OF  AN  ALGEBEAIC  EQUATION.    *         267 

everthing  which  is  of  importance  from  the  algebraic  standpoint  for 
the  equation  considered. 

The  determination  of  the  irreducible  factor  gives  at  once  the 
group  of  the  equation;  in  the  w!  factors 

e  —  (Ui^  x^  +  Hi,  a?2  +  .  .  .  +  Ui^  x„)  * 

of  which  the  Galois  resolvent  is  composed,  we  have  only  to  regard 
the  m's  as  undetermined  quantities,  and  to  form  the  group  of  the  t^'s 
which  permute  the  factors  of  the  irreducible  factor  among  them- 
selves. 

It  must  be  always  borne  in  mind  that  from  the  algebraic  stand- 
point only  those  equations  have  a  special  character,  according  to 
Kronecker  an  affect,  for  which  the  Galois  resolvent  of  the  (n!)^^ 
degree  is  factorable. 

§  227.  On  account  of  the  intimate  connection  between  an  equa- 
tion and  its  group,  we  may  carry  over  the  expressions  "transitive," 
"primitive"  and  "non- primitive,"  "simple"  and  "compound"  from 
the  group  to  the  equation.  Accordingly  we  shall  designate  equations 
as  transitive,  primitive  or  non- primitive,  simple  or  compound,  when 
their  groups  possess  these  several  properties.  Conversely,  we  apply 
the  term  "solvable,"  which  is  taken  from  the  theory  of  equations^ 
also  to  groups,  and  speak  of  solvable  groups  as  those  whose  equations 
are  solvable.  Since,  however,  an  infinite  number  of  equations  belong- 
to  a  single  group,  this  usage. must  be  justified  by  a  proof  that  the 
solution  of  all  the  equations  belonging  to  a  given  group  is  furnished 
by  that  of  a  single  one  among  them.  This  proof  will  be  given 
presently  (Theorem  V). 

In  the  first  place  we  attempt  to  reproduce  the  properties  of  the- 
groups  in  the  form  of  equivalent  algebraical  properties  of  their  equa- 
tions.    We  have  already  (§  150) 

Theorem  II.  If  an  equation  is  irreducible,  its  group  is 
transitive;  conversely,  if  the  group  of  an  equation  is  transitive,  the 
equation  is  irreducible. 

§  228.  To  determine  under  what  form  the  non-primitivity  of  the 
group  reappears  as  a  property  of  the  equation,  we  recur  to  the  treat- 
ment of  those  irreducible  equations  one  root  of  which  was  a  rational 
function  of  another.     The  equation  of  degree  mv  reduced  to  v  equa- 


268  •  THEOEY    OF    SUBSTITUTIONS. 

tions  of  degree  m,  the  coefficients  of  which  were  rationally  expressi- 
ble in  terms  of  the  roots  of  an  equation  of  degree  v  (§  174).  We 
arrive  in  the  present  case  at  a  similar  result. 

Suppose  that  the  group  G  of  the  equation  f{x)  =  0  is  non-prim- 
itive; then  the  roots  of  the  equation  can  be  distributed  into  v  sys- 
tems of  m  roots  each 

such  that  every  substitution  of  the  group  which  converts  one  root 
of  any  system  into  a  root  of  another  system  converts  the  entire 
former  system  into  the  latter.  We  take  now  for  a  resolvent  any 
arbitrary  symmetric  function  of  all  the  roots  of  the  first  system 

1)  Vi'^Six.^x,,,  ..  .x,J 

and  apply  to  S  all   the  substitutions  of  G.     Since  G  is  non-primi 
tive,  the  entire   system  x^-^ ,  a?j2 .  .  .  a?i,„  is  converted  either  into  itself 
or  into  one  of  the  other  systems.     There  are  therefore  only  v  values 
of  y 

Vl    =^>->(^ll5    ^12  J    •    •    •    ^iMijj 

2/2    *^  \'^21  5    ^^22)   •   •   •  '^2w/j 

2/3   — ^  ^-^(^315    ■^32  5    •   •   •  ^3»ij) 


2) 


Consequently  y  is  a  root  of  an  equation  of  degree  v 

3)  v{y)=0, 

the  coefficients  of  which  are  unchanged  by  all  the  substitutions  of 
G,  and  which  are  therefore,  from  Theorem  I,  rationally  known.  "  If 
<p(y):=0  has  been  solved,  i.  e.,  if  all  its  roots  2/15  ?/2?  •  •  •  2/v  ^^^ 
known,  then  all  the  symmetric  functions  of  every  individual  sys- 
tem are  also  known.  For  each  of  these  functions  belongs  to  the 
same  group  as  the  corresponding  y,  and  can  therefore  be  rationally 
expressed  in  terms  of  the  latter  and  of  the  coefficients  of  f{x).  If 
we  denote,   in  particular,  the  elementary  symmetric  functions   of 

*^al  J  '^a2 1  •  •  •  '^arn    t)y 

Si{ya),    Sfya),  .  .  ,  S,„{ya), 

then  the  quantities  Xai ,Xa2,...  x^m  are  the  roots  of  the  equation 

4)  x'^—S.iy^)  x—'-^ Sfya)  x^-'—  .  .  .  ±  5,„  (2/a)  =  0 


THE  GROUP  OF  AN  ALGEBRAIC  EQUATION.  2C9" 

Consequently  f{x)  can  be  obtained  by  eliminating  y  from  3)  and  4)^ 
and  we  have 

V 

a  =  1 

Conversely,  if  we  start  from  the  last  expression,  as  the  result  of 
eliminating  y  from  3)  and  4),  then  the  group  belonging  to  f{x)  =  0 
is  non- primitive,  if  we  assume  that  3)  and  4)  are  irreducible.  For 
we  form  first  a  symmetric  function  of  the  roots  of  4).  This  is 
rational  in  ?/«.     We  denote  it  by  F{ya).     Again  we  form  the  product 

5)  [u-Fiy,)-]  [u-F{y^]  .  .  .  [i^--F(2/.)] 

for  all  the  roots  of  4).  This  product  is  rationally  known;  for  its- 
coefficients  are  symmetric  in  ;//, ,  2/2,  •  •  •  Z/,/,  fincl  are  therefore  ration- 
ally expressible  in  the  coefficients  of  3).  Accordingly  5)  remains 
unchanged  by  all  the  substitutions  of  the  group,  i.  e.,  every  substi- 
tution of  the  group  interchanges  the  linear  factors  of  this  product 
only  among  themselves.  If  therefore  F{ya)  can  be  expressed  in 
terms  of  the  x's  in  only  one  way,  it  follows  that  the  group  converts 
the  symmetric  functions  of  a-a, ,  ^a^?  •  •  •  '^a,v<  io^o  those  of  another 
system.  The  group  is  therefore  non- primitive.  But  if  the  roots  of 
f{x)  are  different  from  one  another,  the  assumption  in  regard  ta 
F{ya)  can  be  realized  by  §  111. 

Theorem  HI.  The  group  of  an  equation  of  degree  mv, 
which  is  obtained  by  the  elimination  of  y  from  the  two  irreducible 
equations 

3)  <p{y)=y^-A,y^-'-\-  . . .  ±  A,  =  0, 

4)  x-—S,{y)x—'^-S,{y)x—'—  .  .  .  ±  S^  =  0 

is  non  primitive ;  and  conversely  every  equation,  the  group  of  which 
is  7ion-pritnitive  is  the  result  of  such  an  elimination. 

§  229-  The  properties  of  an  equation  the  group  of  which  is  com- 
pound do  not  present  themselves  in  so  apparent  a  form  as  in  the  case 
of  the  transitivity  or  non-primitivity  of  the  group.  We  can  however 
replace  the  problem  of  the  solution  of  the  equation  by  another  equiv- 
alent problem  in  which  the  compound  or  the  simple  character  of  the 
group  has  an  easily  observed  effect  on  the  equation  itself. 


270 


THEORY    OF    SUBSTITUTIONS. 


For  this  purpose  we  have  only  to  take  in  the  place  of  the  general 
equation 

6)  f(x)  =  0 
its  Galois  resolvent  equation 

7)  F(i)  =  0. 

F(^)  is  irreducible.  We  have  first  to  examine  more  closely  the 
latter  equation  and  its  properties. 

Given  a  general  equation  6),  there  is  a  linear  function  of  the 
roots  of  6),  formed  with  n  undetermined  parameters 

which  has  n!  values;  so  that  all  the  substitutions  of  the  symmetric 
^roup   G  belonging  to  6)  convert  c^  into  nl  different  values 

The  permutations  among  the  c^,  1^2,  •  •  •  ^ni  produced  by  G  form  a  new 
group  among  the  nl  elements  c,  which  we  denote  by  F.  F  is  simply 
isomorphic  to  G,  and  is  the  group  of  the  equation  7).  F  has  the 
property  that  its  order  is  equal  to  its  degree,  as  appears  either 
from  the  method  of  its  construction,  or  from  the  fact  that  every  c  is 
a  rational  function  of  every  other  one.  The  equation  7),  which  is 
identical  with 

T)  (f-?,)(l-?,)...(l-?,.,)=0. 

therefore  requires  for  its  complete  solution  only  the  determination 
of  a  single  root.     The  solution  of  7)  is  equivalent  to  that  of  6). 

The  question  arises,  how  these  relations  are  modified,  if  we  pass 
from  the  general  equation  6)  to  a  special  equation.  Every  special 
equation  is  characterized  by  a  single  relation  between  the  roots 

9)  <f{3Cj,X2,  .  .  .x,,)  =  0. 

If  (f  belongs  to  a  group  G  of  the  order  r,  then  only  the  substi- 
tutions belonging  to  G  can  be  applied  to  the  roots.  For  if  a  sub- 
stitution were  admitted  which  converted  (p  into 

<fi{x^,  a?2,  •  . .  Xn)  =0, 

where  <fi  is  different  from  (p^  then  all  rational  functions  of  ^ 

a<p(Xi,  a?2,   .   .   .   0Cn)  +  f^<Pl{Xu'^2,   .-.Xn) 


THE  GROUP  OF  AN  ALGEBRAIC  EQUATION.  271 

would  be  rationally  known.     The  rational  domain  thus  determined 
would  be  more  extensive  than  that  derived  from  9?.     Consequently 

9)  would  not  represent  all  the  relations  which  exist  between  the 
roots. 

We  can  now  obtain  a  resolvent  of  our  special  equation  in  either 
of  two  ways.     Either  we  proceed  from 

apply  to  ^1  all  the  r  substitutions  of  G,  obtain 

and  form  the  .resolvent  of  the  r^^  degree 

10)  F,(?)  =  {?-i,)  ( f-,%) .  .  .  (f-f,.)  =  0; 

or  we  proceed  from  the  expression  F{^\  already  given  in  7)  and  7'), 
and  observe  that  F{^)  becomes  reducible  on  the  adjunction  of  9) 
and  that  Fi{B)  is  one  of  the  irreducible  factors.  The  other  fac- 
tors are,  like  F^ ,  of  the  r^^^  degree.  They  differ  from  each  other 
only  in  the  constants  a.  Every  one  of  them  is  obtained  by  multi- 
plying together  all  the  factors  ^  —  Ca,  which  arise  from  the  applica- 
tion of  the  group  6r  to  a  single  one  among  them.  The  group  of 
irj(|)==:0,  regarded  as  a  group  among  the  c's,  is  of  degree  and 
order  r.  It  is  simply  isomorphic  to  the  group  G  of  degree  n  and 
order  r  belonging  to  ^. 

The  groups  of  all  the  factors  Fi{^),  .  .  .  ot  F{^)  therefore  differ 
from  one  another  only  in  the  particular  designation  of  their  elements. 

Theorem  IV,     If  a  special  equation  f(x)  =  0  is   charac- 
terized by  the  family  of 

9)  ^  (o^i ,  a?2 , . . .  x„)  =  0, 

with  a  group  G  of  order  r,  then  the  general  Galois  resolvent  decom- 

poses  into  />  =  —  factors 

10)  *"  F.{S)=0, 

every  one  of  which  can  serve  as  the  Galois  resolvent  of  the  special 
equation.  All  the  roots  of  10)  are  rational  functions  of  every  one 
among  them,  and  in  terms  of  these  all  the  roots  of  f{x)  =0  can  he 
rationally  expressed.  The  transition  from  f{x)  =  0  to  F^{^)  —  0 
has  its  counterpart  in  the  transition  from  G  to  the  simply  isomor- 
phic group  ^  (§  129)  of  F,{$). 


272 


THEORY    OF    SUBSTITUTIONS. 


Since  the  construction  of  10)  depends  only  on'the  group  G,  and 
not  on  the  particular  nature  of  9),  this  same  resolvent  belongs  to  all 
equations  which  are  characterized  by  functions  of  the  same  family 
with  y).  If  one  of  these  equations  has  been  solved,  then  x^,  X2,.  .  .  £c„ 
and  consequently  ^^  are  known.  The  equation  10)  is  therefore  solved, 
and  with  it  every  other  equation  of  this  sort.  We  have  then  the 
proof  of  the  theorem  stated  in  §  226: 

Theorem  V.  Given  an  equation  f{x)  =  0,  the  coefficients  of 
ivhich  belong  to  any  arbitrary  rational  domain,  the  adjunction  of 
either  ^1  =  0  or  ^2  =  0,  where  (p^  and  (p^  belong  to  the  same  family 
of  the  roots  x^,  X2,  .  .  .  x„,  leads,  as  regards  solvability,  to  the  same 
special  equation. 

§  230.  We  have  treated  in  earlier  Chapters  cases  where  such 
relations  between  the  roots  either  were  directly  given  or  were  easily 
recognized  as  involved  in  the  data.  Frequently,  however,  the 
conditions  are  such  that,  instead  of  a  known  function,  4'{^\  ,oc2...  x„) 
being  directly  designated  as  adjoined,  (J>  presents  itself  implicitly  as 
a  root  of  an  equation  which  is  regarded  as  solvable.  For  example, 
in  the.  problem  of  the  algebraic  solution  of  equations  the  auxiliary 
equation  is  of  the  simple  form 

y^  —  ^(^1,  X2,  .  .  .  X,)  =  0. 

Here  y  is  regarded  as  known,  i.  e.,  we  extend  the  rational  domain 
of  f{x)  =  0  by  adjoining  to  it  every  rational  function  of  the  roots 
of  which  any  power  belongs  to  the  domain.  The  actual  solution  of 
the  auxiliary  equations  does  not  enter  into  consideration. 

It  is  a  natural  step,  when  an  irreducible  auxiliary  equation  is- 
regarded  as  solvable,  to  adjoin  not  one  of  roots  4'-,  but  all  of  its 
roots  to  the  domain  of  f{x)  —  0.  These  roots  are  the  different  val- 
ues which  4'{^\',  X2,  .  .  .  x^  assumes  within  the  rational  domain.  For 
to  find  the  auxiliary  equation  which  is  satisfied  by  4\  we  apply  to 
4\  all  the  r  substitutions  of  the  group  G  and  obtain,  for  example, 
m  distinct  values 

11)  ^'l,V''2,V''3,...V''.. 

The  symmetric  functions  of  these  values,  and  therefore  the  coeffi- 
cients of  the  equation 


THE  GROUP  OF  AN  ALGEBRAIC  EQUATION.  273 

12)  g{4')  =  {4'—4\)(4'—4'2)  •  •  •  {<l'—4>m)  =  ^ 

are  known  within  the  rational  domain  of  f{x)  =  0,  and  12)  is  the 

required  auxiliary  equation,  the  solution  of  which  is  regarded  as 

known. 

Now  given  the  equation  f{x)  =  0,  characterized  by  the  group  G, 
or  by  any  function  <p{x^,X2,  .  ..  x^)  belonging  to  G,  we  adjoin  to 
it  all  the  roots  of  12),  or,  what  amounts  to  the  same  thing,  a  linear 
combination  of  these  m  roots 

/  =  «101  +  «2</'2  +   .   •   .   +  «m  4'm, 

where  the  a's  are  undetermined  constants.  The  question  then  arises, 
what  the  group  of  /(a?)  =  0  becomes  under  the  new  conditions. 

The  adjoined  family  of  functions  was  originally  that  of  (f .  Now 
it  is  that  of 

<P  +  X  =  'P  +  ^l4'l-\-  ^24'2+   •   .  •   -\-(^m4'm' 

The  group  was  originally  G.  Now  it  is  that  subgroup  of  G,  which, 
is  also  contained  in  all  the  groups 

-ff  1  J    H2  ,   .   .   .   -ff m  ) 

<^f  4'i  1  4'2i  •  •  •  4'm •  Suppose  that  K  is  the  greatest  common  subgroup 
of  these  m  groups.     Then  K  belongs  to  the  function  /- 

If  now  we  apply  all  the  substitutions  of  G  to  the  series  4'\ »  02  ?  -.  4'mY 
the  result  is  in  every  case  the  same  series  in  a  new  order;  for 
v''ij  <r'25  .  •  •  4'm  are  all  the  values  which  G  produces  from  v''i-  Conse- 
quently the  series  H^,  H2,  .  .  .  H^  is  also  reproduced  by  transforma- 
tion with  respect  to  G;  and  K  is  therefore  unchanged  by  transform- 
ation with  respect  to  G.     We  have  then 

G-^KG  =  K. 

Again  we  denote  by  -T  the  greatest  subgroup  of  G  which  is  con- 
tained in  K.  r  therefore  belongs  to  ^+/,  and  accordingly  char- 
acterizes the  family  which  belongs  to  f{x)^=0  after  the  adjunction 
of  all  the  roots  of  12).  T,  like  K,  is  also  commutative  ivith  G;  for 
on  transforming  F  with  respect  to  G,  the  result  must  belong  to  both 
G  and  K,  and  is  therefore  F  itself.  F  is,  then,  a  self- conjugate  sub- 
group of  G,  and  in  fact  is  the  most  comprehensive  of  those  which 
are  also  common  to  H^,  H2,  . .  .  H^. 

If  /'  does  not  reduce  to  the  identical  operation,  G  is  a  compound 
18 


274  THEORY    OF    SUBSTITUTIONS. 

group.  If  G  is  simple,  T  is  necessarily  identity,  and  the  group  of 
f{x)  —  0  is  reduced  by  the  solution  of  12)  to  1,  i.  e.,  after  the 
solution  of  12)  all  the  roots  of  f{x)  =  0  are  known;  or,  in  other 
words,  the  solution  of  12)  furnishes  that  oif{x)  —  0  also.  We  have 
then  the  following 

Theorem  VI.  Given  any  arbitrary  equation  f(x)  =  0  with 
the  group  G,  if  we  adjoin  to  it  all  the  roots 

11)  01,02,   03,   .   ..0m 

of  an  irreducible  equation  of  the  m*^  degree 
12)  ^(0)^0*"— A^'"-^+...  =0, 

the  coefficients  of  which  are  rational  in  the  rational  domain  of  f{x), 
and  the  roots  rational  functions  of  x^,  X2,  .  .  .  x„,  then  G  reduces  to 
the  largest  self- conjugate  subgroup  F  of  G  which  leaves  4'i,  02,  • .  •  0m 
all  unchanged.  If  G  is  a  simple  group,  Jr=  1.  O^ily  in  case  G  is 
compound  is  it  possible  by  the  solution  of  an  auxiliary  equation  to 
reduce  the  group  to  a  subgroup  different  from  identity,  and  conse- 
quently to  divide  the  Galois  resolvent  equation  into  non-linear  fac- 
tors. 

§  231.  We  consider  these  results  for  a  moment.  If  the  general 
equation  of  the  n^'^  degree  f{x)—0  is  given,  the  corresponding 
group  G  is  of  order  r  —  n\.  This  group  is  compound,  the  only 
actual  self-conjugate'subgroup  being  the  alternating  group  (§92). 
If  we^take  for  a  resolvent 

01  =V^, 
"where  A  denotes,  as^usual,  the  discriminant  oif{x),  then  the  resolv- 
ent equation  becomes 
12')  4,^—A  =  0, 

and  r  is  the  alternating  group.  After  adjunction  of  the  two  roots 
of  12')  the  previously  irreducible  Galois  resolvent  equation  divides 
into  two  [conjugate  factors  of  degree  Jw!,  and  only  such  substitu- 
tions can  be  applied  to  the  resolvent 

as  leave  V  J  unchanged  and  therefore  belong  to  the  alternating 
group. 

For  n  >  4  the  alternating  group  is  simple.     If  there  is  an  m- 


THE  GROUP  OF  AN  ALGEBRAIC  EQUATION.  275 

valued  resolvent  v'',  its  values  4\^  <,''2,  .  .  .4>m  are  obtained  by  the 
solution  of  an  equation  of  the  w}^  degree.  On  the  adjunction  of 
these  values,  or  of 

the  group  of  the  given  equation  reduces,  by  Theorem  VI,  to  the 
identical  substitution.  The  equation  /(x)  =  0  is  therefore  solved; 
for  all  functions  are  known  which  belong  to  the  group  1  or  to  any 
other  group.  The  investigations  of  Chapter  VI  show,  however,  that 
no  reduction  of  the  degree  of  the  equation  to  be  solved  can  be 
effected  in  this  way,  since  if  n  >  4,  the  number  m  of  the  values  of  </', 
if  it  exceeds  2,  is  greater  than  n  or  equal  to  n.  In  the  latter  case, 
if  n=p6,  the  function  4'  is  always  symmetric  in  n  —  1  elements,  so 
that  we  can  take  directly  4'\  —  ^i ,  and  the  resolvent  equation  is  iden- 
tical with  the  original  f{x)  =  0. 

Theorem  VII.  The  general  equation  of  the  n''*  degree 
(n  >  4)  is  solved^  as  soon  as  any  arbitrary  resolvent  equation  of  a 
degree  higher  than  the  second  is  solved.  There  are,  however,  no 
resolvent  equations  the  degree  of  ivhich  is  greater  than  2  and  less 
than  n.  Moreover,  if  n==6,  there  is  no  resolvent  equation  of  the 
n*^  degree  essentially  different  from  f(x)  =  0.  For  n  =  6  there  is 
a  distinct  resolvent  equation  of  degree  6. 

One  other  result  of  our  earlier  investigations,  as  reinterpreted 
from  the  present  point  of  view,  may  be  added  here: 

Theorem  VII J.  The  general  equation  of  the  fifth  degree 
■has  a  resolvent  equation  of  the  sixth  degree. 

§  232.     We  return  now,  from  the  incidental  results  of  the  pre- 
ceding   Section,  to    Theorem  VI,  and  examine  the    group  of   the 
equation 
12)  flr(9^)  =  (0  — 00  {4'  — 4',)  ...  (0—  0.)  =0, 

the  roots  4\  ?  </'2  ?  •  •  •  4'm  of  which  were  all  adjoined  to  the  equation 
/(a-)  =  0. 

The  order  of  the  group  of  12)  is  most  easily  found  from  the  fact 
that  it  is  equal  to  the  degree  of  the  irreducible  equation  of  which 

is  a  root.     We  must  therefore  apply  to  w  all  the  r  substitutions  of 


276  THEOEY    OF    SUBSTITUTIONS. 

G.  The  values  thus  obtained  may  partly  coincide.  The  number  of 
distinct  values  gives  the  order  of  the  group  of  12).  Now  if  we  retain 
the  designations  of  Theorem  VI,  r  includes  all  substitutions  of 
iTi ,  a-o ,  .  .  .  a:?„  which  leaves  all  the  0's  unchanged.  Suppose  that 
the  order  of  F  is  r'.     Then  the  required  number  is  \i  zzur-.r'. 

From  this  we  perceive  that,  if  the  group  G  is  simple  and  F  ac- 
cordingly of  order  ?-'  =  !,  the  order  v  of  the  group  of  every  resolvent 
equation  is  the  same  as  that  of  f{x)  —  0,  so  that  no  simplification 
can  be  effected  in  this  way. 

We  actually  obtain  the  group  of  12)  by  the  consideration  that 
it  contains  all  and  only  those  substitutions  among  the  x's  which 
do  not  alter  the  nature  of  f{x)  =  0.     If  therefore  we  apply  to 

<J'l,  ^''2,  V''3,   .   .  '</'m 

all  the  substitutions  of  G,  the  resulting  permutations  of  the  0's  form 
the  group  required.  All  the  r  substitutions  thus  obtained  are  not 
however  necessarily  different ;  for  all  the  substitutions  of  /'  leave  all 
the  elements  v''  unchanged.  From  this,  again,  it  follows  that  the 
order  of  the  group  K  of  12)  is  v  =  r:r\  In  the  same  way  we  recog- 
nize that  K  is  (1 — r)  fold- isomorphic  to  G.  With  the  notation  of 
§  86,  K  is  the  quotient  of  G  and  F;  K  =  G:r, 

Tlieorem  IX.  If  the  group  G  of  f{x)  =  0  is  of  order  r, 
and  contains  a  self-conjugate  subgroup  F  of  order  r',  and  if  G 
reduces  to  F  on  the  adjunction  of  all  the  roots  11)  of 

12)  fir(v'')  =  0, 

then  the  group  K  of  the  latter  equation  is  of  order  v  =r :  r'.     K  is 

the  quotient  of  G  and  F  and  is  (1  —  r)'fold  isomorphic  to  G. 

By  a  proper  choice  of  the  resolvent  v''  we  can  give  the  equation 
12)  a  very  special  character. 

We  choose  as  a  resolvent  a  function  y  belonging  to  the  self-con- 
jugate subgroup  /'.  Then  /  is  a  root  of  an  equation  of  degree 
vz^r.r',  all  the  roots  of  which  are  rationally  expressible  in  terms  of 
anyone  among  them;  for  /u  ;^2>  •••/>'  ^,11  belong  to  the  same  group 
r,  (§  109,  Theorem  VIII).  The  group  of  12)  is  therefore  a  group  ii\ 
for  it  is  transitive,  since  g{x)  is  irreducible.     We  have  therefore 

Theorem  X.     If  the  group  G  of  the  equation  f{x)  —  0  is 


THE  GROUP  OF  AN  ALGEBRAIC  EQUATION.  277 

of  order  r,  and  contaiyis  a  self- conjugate  subgroup  F  of  order  r', 
and  if  y^  is  a  function  of  the  roots  x^,  X2,  .  .  .  x^,  belonging  to  F, 
then  an  irreducible  resolvent  equation  of  degree  v  =  r:7'' 

<ian  be  constructed,  the  roots  of  ivhich  are  all  rational  functions  of 
a  single  one  among  them,  and  which  possesses  the  property  that  the 
adjunction  of  one  of  its  roots  to  f{x)  =  0  reduces  the  group  G  to  F. 

§  233.  Theorem  XI.  //  F  is  a  maximal  self- conjugate 
subgroup  of  G,  then  the  group  of  h{x)  =  0  is  a  transitive,  simple 
group.  Conversely,  if  F  is  not  a  most  extensive  self -conjugate  sub- 
group of  G,  then  the  group  of  h{/)  —  0  is  compound. 

We  denote  the  group  of  h{x)  =  0  by  G'.  Its  order  is  \>  =ir\r'. 
We  assume  that  G'  contains  a  self- conjugate  subgroup  /'',  of  order  r^. 
From  Theorem  IX  G'  is  r'-fold  isomorphic  to  G.  From  the  results 
of  §  73  it  follows  that  the  subgroup  J  of  G,  which  corresponds 
to  the  group  F',  is  a  self- conjugate  subgroup  of  G  and  is  of  order 
v'r'.  J  is,  then,  like  F,  a  self- conjugate  subgroup  of  G,  and 
their  orders  are  respectively  i-'  r'  and  r'.  We  show  that  F  is  con- 
tained in  J.  This  follows  directly  from  the  construction  of  G'  (§  232), 
in  accordance  with  which  the  substitution  1  of  G'  corresponds  to  all 
the  substitutions  of  G  which  leave  the  series  11)  unaltered.  F  in.  G 
therefore  corresponds  to  the  one  substitution  1  of  G'.  Accordingly 
if  G'  is  compound,  then  F  is  not  a  maximal  self -conjugate  subgroup 
of  G. 

The  converse  theorem  is  similarly  proved  from  the  properties  of 
isomorphic  groups. 

In  these  last  investigations  we  have  dealt  throughout  with  the 
group  of  the  equation,  but  never  with  the  particular  values  of  the 
coefficients.  If  therefore  two  equations  of  degree  7i  have  the  same 
group,  the  reductions  of  the  Theorem  X  are  entirely  independent  of 
the  coefficients  of  the  equations.  The  coefficients  of  h{x)  will  of 
course  be  different  in  the  two  cases,  but  the  different  equations 
h{x)  =  0  all  have  the  same  group,  and  every  root  of  any  one  of  these 
equations  is  a  rational  function  of  every  one  of  its  roots.  This  com- 
mon property  relative  to  reduction,  which  holds  also  for  the  further 


278  THEORY    OF    SUBSTITUTIONS. 

investigations  of  the  present  Chapter,  is  the  chief  reason  for  the 
collection  of  all  equations  belonging  to  the  same  group  into  a  family. 

§  234.  We  observe  f  urtlier  that  with  every  reduction  of  the  group 
there  goes  a  decomposition  of  the  Galois  resolvent  equation,  v^hile 
the  equation  f{pc)  =■  0  need  not  resolve  into  factors. 

Collecting  the  preceding  results  we  have  the  following 

Theorem  XII.  If  the  group  G  of  an  equation  f(x)  =  0  is 
compound,  and  if 

G,  Gi,   G2,  .  .  .  Gy,   1 
is  a  series  of  composition  belonging  to  G,   so  that  every  one  of  the 
groups  Gi,  G21  .  .  .  Gy,  1  is  a  maximal  self- conjugate  subgroup  of 
the  preceding  one,  further  if  the  order  of  the  several  groups  of  the 
series  are 

then  the  problem  of  the  solution  of  f{x)  =  0  can  be  reduced  as  foU 
lows.  We  have  to  solve  in  order  one  equation  of  each  of  the  de- 
grees 

r      r^      r^  iV-_i 

^i'  n'  rs'  •••  r,  '  '''" 
the  coefficients  of  which  are  rational  in  the  rational  domain  deter- 
mined by  the  sohdion  of  the  preceding  equation.  These  equations 
are  irreducible  and  simple,  and  of  such  a  character  that  all  the 
roots  of  any  one  of  them  are  expressible  rationally  in  terms  of  any 
root  of  the  same  equation.  The  orders  of  the  groups  of  the  equa- 
tions are  respectively 


^     ^      ^2  ^V-i 

n  '     ^2 '      ^3  '   '  "       '^v      ' 


The  groups  are  the  quotients 

G  :  Gi,      Gi'.  G2 5      G2 '.  G3 5  .  .  .  Gy _  1 :  G^, ,      Gy :  1. 
The  equations  being  solved,  the  Galois  resolvent  equation,  ichich  was 
originally  irreducible  and  of  degree  r,  breaks  up  successively  into 


/V»  /V»  «J^  ^ 


' 1  '2  '3 

factors.     After  the  last  operation  f{x)  —  0  is  therefore  completely 
solved 


THE  GEOUP  OF  AN  ALGEBRAIC  EQUATION.  279 

§  235.  The  composition  of  the  group  G  of  an  equation  f{x)  =  0 
is  therefore  reflected  in  the  resolution  of  the  Galois  resolvent  equa- 
tion into  factors.  We  turn  our  attention  for  a  moment  to  the  ques- 
tion, when  a  resolution  of  the  equation  f{x)  =  0  itself  occurs.  It  is 
readily  seen  that,  in  passing  from  Ga  to  Ga+i  in  the  series  of  com- 
position of  G,  a  separation  of  f{x)  into  factors  can  only  occur  when 
Ga-^-i  does  not  connect  all  the  elements  transitively  which  are  con- 
nected transitively  by  Ga.  The  resulting  relations  are  determined 
by  §  71.  Ga  is  non-primitive  in  respect  to  the  transitively  con- 
nected elements  which  Ga+i  separates  into  intransitive  systems. 

Starting  from  G,  with  an  irreducible  f(x)  =  0,  suppose  now  that 
Gi,  G^,  .  .  .  Ga  are  transitive,  but  that  Ga+i  is  intransitive,  so  that 
by  §  71  Ga  is  non- primitive.  Then  at  this  point  f(x)  separates 
into  as  many  factors  as  there  are  systems  of  intransitivity  in  Ga  +  n 
But  (again  from  §71),  all  the  elements  occur  in  Ga+i-  We 
arrange,  then,  the  substitutions  of  Ga  in  a  table  based  on  the  sys- 
tems of  intransitivity  of  6ra  +  i-  Suppose  that  there  are  //  such  sys- 
tems, so  that  f{x)  divides  into  fj.  factors.  Then  we  take  for  the  first 
line  of  the  table  all  and  only  those  substitutions  of  Ga,  which  do 
not  convert  the  elements  of  the  first  system  of  intransitivity  into 
those  of  another  system.  The  substitutions  of  this  line  form  a 
group,  which  is  contained  in  Ga  as  a  subgroup.  Its  order  is  there- 
fore kVa+i.  The  second  line  of  the  table  consists  of  all  the  substi- 
tutions of  Ga  which  convert  the  first  system  of  intransitivity  into 
the  second.  The  number  of  these  is  also  kVa  +  i.  There  are  n  such 
lines,  and  they  include  all  the  substitutions  of  G%.     Consequently 


kr, 


a  +  l 


i.  e.,  the  number  ix  of  the  factors  into  which  f(x)  divides  is  a  divi- 

r 
sor  of  the  number  of  the  factors  into  which  the  Galois  resolvr 

''"a+  1 

ent  equation  divides  at  the  same  time.  A  similar  result  obviously 
occurs  in  every  later  decomposition. 

The  decomposition  can  therefore  only  take  place  according  to  the 
scheme  of  Theorem  III.  The  several  irreducible  factors  are  all  of 
the  same  order. 

§  236.     Thus  far  we  have  adjoined  to  the  given  equation  /(a?)  =  0 


280  THEORY    OF    SUBSTITUTIONS. 

the  root  4'  of  a  second  irreducible  equation  only  when  the  <^'''s  were 
rational  functions  bi  x^^x^,  .  .  .x^.     This  seems  a  strong  limitation. 
We  will  therefore  now  adjoin  to  the  equation  f{x)  =  0  all  the  roots 
of  an  irreducible  equation 
18)  g{z)  =  Q 

without  making  this  special  assumption.  The  only  case  of  interest 
is  of  course  that  in  which  the  adjunction  produces  a  reduction  in  the 
group  G  of  f{x)  =  0. 

In  the  first  instance  we  adjoin  only  a  single  root  z^  of  g{z)  —  0. 
Suppose  that  G  then  reduces  to  its  subgroup  H^.  If  the  rational 
function  <Pi{x^,  ^2,  .  .  .  x,)  belongs  to  H^^  then  the  same  reduction 
of .  G  can  be  produced  by  adjoining  cp-^  instead  of  z^ .  Suppose  that 
under  the  operation  of  G,  the  function  <p^  takes  the  conjugate  values 
^\i  ^2^,  •  •  '  ^mi  with  the  groups  H^,  H^,  ...  H,„  respectively.  These 
values  satisfy  an  irreducible  equation 

k{<p)  =  {<p  —  <p,)  (^—^2)..  .{<p  —  <p„,)  =  0. 
The  adjunction  of  z^  to  the  rational  domain  has,  by  the  mediation 
of  Hi ,  made  ^1  rational,  so  that  ^j  is  a  rational  function  of  z^ 

<Pi{xi,x^, .  ..  x„)  =  4iz,). 

It  appears  therefore  that,  in  order  that  the  adjunction  of  z^  may  pro- 
duce a  reduction  of  G,  it  is  necessary  and  sufficient  that  there  should 
be  a  rational  non- symmetric  function  of  the  roots  v'i(^n^2?  •  •  •  ^n) 
which  is  rationally  expressible  in  terms  of  2;i . 

Suppose  that  the  roots  of  the  irreducible  equation  g(z)  =  0  are 
Zi,  Zo,  .  .  .  Zfj,,  and  tllat  its  group  is  F.     Since  A:[v''(2;)]  =  0  is  satisfied 

by  2^1,  all  the 

<p{z;)     (i=l,2,3,.../.) 

are  roots  of  k(<p)  =  0;  that  is  (/'(z^),  4'{z^),  .  .  .  4'{Ztj)  are  the  conjugate 
Talues  of  (pi .     On  the  other  hand  the  coefficients  of  the  product 

are  symmetric  functions  of  the  roots  of  g{z)  —  0  and  are  therefore 
rationally  known;  and  the  equation  k^  —  ^  has  all  its  roots  in  com- 
mon with  A:  =  0.     Consequently  k^i^p)  is  a  power  of  k{(p^) 

k,{<p)  =  k^{<p), 
and  the  //  values  ^H^i)?  ^''fe)?  •  •  •  ^''(^m)  coincide  in  sets  of  q  each. 


THE  GROUP  OF  AN  ALGEBRAIC  EQUATION.  281 

^ith  a  slight  change  in  the  notation  for  the  z's  we  can  therefore 
write 


Since  (/'(Za^y^)  —  <p{zp^y^)  =  0,  this  quantity  is  rationally  known. 
It  is  therefore  unchanged  by  all  the  substitutions  of  1\  i.  e.,  T  inter- 
changes the  lines  of  %),  and  therefore  gives  rise  to  a  group  2'  of  the 
elements  <p  which  is  isomorphic  to  F.  To  the  substitution  1  in  T 
correspond  in  F  the  substitutions  of  the  subgroup  ^  of  order  d^  which 
only  interchange  z\,z'2,  .  .  .  z\  among  themselves,  z'\,  z'\,  .  .  .  z''^ 
;among  themselves,  and  so  on.    7'  and  T  are  (1  — di)-fold  isomorphic. 

If  we  coordinate  all  the  substitutions  of  G  and  T  which  leave  (fi 
unchanged,  and  again  one  substitution  each  from  G  and  T  which 
<5on verts  <pi  into  9"2,  one  which  converts  ^i  into  ^3,  and  so  on,  an 
isomorphism  is  also  established  between  G  and  T.  To  the  substitu- 
tion 1  in  r  correspond  in  G  the  substitutions  of  the  subgroup  D  of 
order  d  which  is  the  maximal  common  subgroup  of  H^,  H2,  .  .  .  H^. 

Accordingly  G  and  F  are  also  isomorphic,  and  in  fact  their  iso- 
morphism is  (d  —  di)-fold,  as  shown  by  the  preceding  considerations, 
and  again  (r — ri)-fold,  as  appears  from  the  orders  of  G  and  F. 
'Consequently 

r:r,  =  ci:d,;     -  =  ^. 

If  now  we  adjoin  to  the  equation  f{x)  —  0  all  the  roots  of 
g (z)  =  0,  then  (pi,  <p2,  .  -  .  <Pm  are  rationally  known.  G  reduces  to  the 
subgroup  D  of  order  d  common  to  the  groups  H^,  H^,  .  .  .  H^.  To 
J)  belongs  the  function 

•and  every  function  of  X]^,  X2,  .  .  .  Xn  which  can  be  rationally  expressed 
in  terms  of  z^,  Z2,  .  .  .  Zf^  belongs  to  the  family  of  p  or  to  an  inclu- 
ded family.  For  every  such  function  is  rationally  known,  as  soon 
as  the  Zi,  Z2,  .  .  .  Zij,  are  adjoined  to  the  equation  f{x)  =  0. 

Conversely,  if  we  adjoin  to  the  equation  g(z)=0  all  the  roots 
of  f{x)  =  0,  it  follows  by  the  same  reasoning  that  there  is  a  function 

l-o)  u,^{zi,  Z2,,..z^)  =po{ocu  X2,...  x^\ 


% 


282  THEORY    OF    SUBSTITUTIONS. 

such  that  every  function  of  z^,  Z2,  .  .  .  Zfj,  which  can  be  rationally 
expressed  in  terms  of  x^,  X2,  .  .  .  x„  belongs  to  the  family  of  o)q  or  to- 
an  included  family. 

Since  now  p^  is  rational  in  the  :s's,  it  follows  from  the  above  prop- 
erty that 

16)  />o  =  -K(/>), 

where  i^  is  a  rational  function;  and  since  w  is  rational  in  the  a?'s,  it 
follows  that 

17)  (O  —  Ro{<0o) 

or,  which  is  the  same  thing, 

17')  p^RoiPo)' 

From  16)  and  17')  it  follows  that  p  and  Pq  belong  to  the  same 
family.  The  adjunction  of  all  the  roots  of  /=0  to  g  =  0  there- 
fore gives  rise  to  the  same  rational  domain  as  the  adjunction  of  all 
the  roots  of  ^  =  0  to  /  =  0. 

It  is  obvious  at  once  that  the  first  adjunction,  since  it  made 
9ii  ^2y  '  '  '  <Pm  rational,  also  furnished  m,  so  that  F  reduces  to  J . 
But  the  proof  just  given  was  necessary  to  exclude  the  possibility  of 
any  further  reduction. 

If  we  write  ~:  =  -^='^,  it  follows  that  if  the  second  adjunction 
d      di 

reduces  the  order  r  of  (r  to  its  v^^  part,  then  the  first  adjunction. 

also  reduces  the  order  r^  of  F  to  its  v^^  part. 

Theorem  XI LI.  The  effect  of  the  adjunction  of  all  the 
roots  of  any  arbitrary  equation  13)  on  the  reduction  of  the  gr-oup 
G  of  f{x)  =  0  can  be  equally  well  produced  by  the  adjunction  of 
all  the  roots  of  an  equation  12)  which  is  satisfied  by  rational  func- 
tions of  X^,  X2,  ,  .  .  Xn. 

In  spite  of  removal  of  apparent  limitations,  we  have  therefore 
not  departed  from  the  earlier  conditions,  where  only  the  adjunction, 
of  rational  functions  of  the  roots  was  admitted. 

§  237.    Theorem  XIV.    // 

f{x)  =  0,  g{z)=^0 

are  two  equations,  the  coefficients  of  which  belong  to  the  same 
rational  domain,  and  which  are  of  such  a  nature  that  the  solution 


THE  GKOUP  OF  AN  ALGEBKAIC  EQUATION. 


285 


of  the  second  and  the  adjunction  of  all  its  roots  to  the  first  reduces 
the  group  of  f{x)  =  0  to  a  self- conjugate  subgroup  contained  in  it 
of  an  order  v  times  as  small,  then  conversely  the  solution  of  the 
first  equation  reduces  the  group  of  the  second,  to  its  v**^  part.  The 
group  of  f{x)  =  0,  like  that  of  g  (z)  =  0,  is  compound,  and  v  is  a 
factor  of  compositio7i.  Those  rational  functions  of  the  roots  of 
one  of  the  two  equations,  by  which  the  same  reduction  of  its  group 
is  accomplished  as  by  the  solution  of  the  other  equation  are  rational 
in  the  roots  of  the  latter. 

As* we  see,  the  group  of  f(x)  =  0  can  be  reduced  by  the  solution 
of  an  equation  g  (z)  =  0,  although  the  roots  of  the  latter  are  not 
rational  functions  of  Xi,X2,  .  .  .  x„.  It  is  only  necessary  that  there 
should  be  rational  functions  of  Zj,Z2,  .  .  .  Zf^  which  are  also  rational 
functions  of  x^,  X2,  .  .  .  x^.  . 

From  the  preceding  Theorem  follow  at  once  the  Corollaries 

Corollary  I.  If  the  group  G  of  the  equation  f{x)  =  0  is 
simple,  the  equation  can  only  be  solved  by  the  aid  of  equations  with 
groups  the  orders  of  which  are  multiples  of  the  order  of  G. 

For  since  G  reduces  to  1,  the  v  of  Theorem  XIV  must  be  taken 
equal  to  the  order  of  G. 

Corollary  II.  //  the  group  G  of  f{x)  —  0  can  be  reduced 
by  the  solution  of  a  simple  equation  g (z)  =  0,  then  z^,  Z2, .  .  .  z,^  are 
rational  functions  of  the  roots  of  f(x)  =  0. 

For  in  this  case  v  is  equal  to  the  order  of  the  group  of  g  (z)  —  0. 
After  the  reduction  this  is  equal  1.     Consequently 

il'\  —a^z^  +  a^z^  +  .  .  .  -f  o.^z^,  =  ir\{x,  ,X2,...  Xn), 

where  ^I'\  is  the  Galois  resolvent  of  g{z)  =  0. 

Corollary  III.  If  the  adjunction  of  the  roots  of  g(z)  =  0 
is  to  produce  a  reduction  of  the  group  of  f(x)  =  0,  then  the  orders 
T  and  rj   cannot  be  prime  to  each  other. 

The  preceding  Sections  contain  a  proof  and  an  extension,  rest- 
ing entirely  on  considerations  belonging  to  the  theory  of  substitu- 
tions, of  Theorem  II,  §  216,  where  the  subject  was  treated  purely 
arithmetically.     For  if  we  retain  the  notation  of  §  216,  it  follows 


284:  THBOKY    OF    SUBSTITUTIONS. 

that  since  V^"  =  F^  is  a  simple  equation,  Vy  is  a  rational  function 
of  the  roots  of  f{x)  =  0,  and  so  on. 

The  proof  of  the  impossibility  of  the  algebraic  solution  of  the 
equations  of  higher  degree  might  therefore  be  based  on  the  present 
considerations. 

§238.  As  the  adjunction  of  the  roots  of  a  new  equation  g(z)  =  0 
to  f(x)  =  0  leads  to  nothing  more  than  the  adjunction  of  rational 
functions  of  the  roots  of  f{x)  =  0,  so  no  new  result  is  obtained,  if 
the  roots  of  both  equations  are  connected  by  a  rational  relation. 
We  prove 

Theorem  XV.     If 

f{x)  =  0,  g{z)  =  0 

are  two  irreducible  equations,  the  roots  of  which  are  connected  with 
each  other  by  rational  relations 

the  latter  can  all  be  obtained  from  a  single  relation  of  the  form 

ir'\\pC\  ,  X2,   .   .   .   X^-t)  =  ^(^1  )  2^2?   •   •    •   ^M/? 

in  ivhich  the  roots  of  the  two  equations  are  separated. 

For,  if  we  denote  the  corresponding  Galois  resolvents  by  c  and  C 
and  the  irreducible  resolvent  equatioQs  of  f{x)  =  0  and  (/  (2;)  =  0  by 

the  degrees  r  and  r'  of  F  and  G  are  equal  to  the  orders  of  the 
respective  groups. 

Now  x^^Xo,  .  •  .  x^  can  be  rationally  expressed  in  terms  of  c,  and 
z^^z^i  .  .  .  Zf^  in  terms  of  C,  so  that 

(p{x,,X2,  .  .  .  ^„;  2^1,  2^2,  .  .  .  2^)  =  '^(--,  0  =  0. 

The  expression  (P  can  be  so  reduced  by  the  aid  of  i^'  =  0  and  (x  =  0 
that  its  degree  becomes  less  than  r  in  c  and  less  than  r'  in  C.  Then 
the  two  equations 

have  a  common  root.  Consequently,  if  we  add  C  to  the  rational 
domain,  the  resolvent  F{:)  becomes  reducible,  since  otherwise  the 
irreducible  equation  of  the  r*^  degree  would  have  a  root  in  common 


THE    GROUP    or    AN    ALGEBRAIC    EQUATION.  285 

with  an  equation  of  a  degree  less  than  r.  The  only  exception 
occurs  when  <^(r,  C)  is  identically  0. 

If  this  does  not  happen,  the  adjunction  of  all  the  roots  of  g(z)=^i} 
or  that  of  ^  breaks  up  the  resolvent  of  f{oo)  =  0  into  factors,  and  we 
have  therefore  the  case  of  the  last  Section.  We  can  effect  the  same 
reduction  by  the  adjunction  of  a  rational  function  /  of.  x^,  X2,  ...  oc^, 
and  we  have 

X{X,,X2,  .  .  .  X,,)  =  C/'(Z,,  Z^,...  Zfj). 

If  several  such  relations  exist,  they  can  all  be  deduced  from  one  and 
the  same  equation.  The  latter  can  be  easily  found,  if  we  select  a 
function  /  such  that  all  the  others  belong  to  an  included  family. 

On  the  otner  hand  if  '/'(,-,  ^)  is  identically  0,  it  follows  that  the 
coefficients  in  the  polynomial  </'(r,  t)  arranged  according  to  powers 
of  r  vanish,  so  that  we  have  equations  of  the  form 

and  similarly,  if  0{^,  t)  is  arranged  in  powers  of  ^, 

These  equations  can  actually  make  </*  -  0.  But  this  amounts 
to  only  an  apparent,  not  an  actual  dependence  of  the  roots  of 
f(x)  ---  0  and  g  (z)  =  0.  The  function  •/2  —  0  belongs  to  the  group 
of  g{z)  —  0,  and  ^\{xx^  x^^  .  .  .x^  belongs  to  the  group  of  f{x)  —  0. 


CHAPTER  XV. 


ALGEBRAICALLY  SOLVABLE  EQUATIONS. 
§  239.     In  §  234  we  have  established  the  following  theorem.     If 
the  group  G  of  the  equation  f{x)  =  0  has  the  series  of  composition 

1)  G,  G-i,  G2,  .  .  .  Gy,   1, 

and  if  the  orders  of  these  several  groups  are 

'*  ?  ^n  ^2 »  •  •  •  ^v  ?   J  ? 
then  the  solution  of  f(x)  =  0  can  be  effected  by  solving  a  series  of 
simple,  irreducible  equations  of  degrees 

I      ^1      ^  n--i 

'  I  '2  '3  '  Pj 

the  first  of  which  has  for  its  coefficients  functions  belonging  to  G 
^nd  for  its  roots  functions  belonging  to  (9, ,  the  second  coefficients 
I)elonging  to  G^  and  roots  belonging  to  Go,  and  so  on.  All  these 
•equations 

*  /i  =  0,     X2  =  0,  .  .  .x^,  =  0,     x^.  +  ^  =  0 

iiave  the  property  that  the  roots  of  any  one  of  them  are  all  rational 
functions  of  one  another,  so  that  the  order  of  the  corresponding 
:group  is  equal  to  its  degree  i.  e.,  the  group  is  of  the  type  i2(§  129). 
We  have  now  to  examine  under  what  circumstances  all  these 
•equations  Xi,  =  0  become  binomial  equations  of  order  p^ 

X^^zn  —  H^  =  0, 

where  Ifx  is  rational  in  the  quantities  belonging  to  the  family  of 
6rA  _  1 .     In  other  words,  we  have  to  determine  the  necessary  and 
sufficient  condition  that  f{x)  =  0  shall  be  algebraically  solrable. 
For  this  result  it  is  necessary  that  the  factors  of  composition 

■M  M  n^  . 

—,—,-,...  should  all  be  prime  numbers,  Pi,P2?P35  •  •  •    For  these 

Ti      7-2     n 

quotients  give  the  degrees  of  the  equations  /i  =  0,  X2  =  ^i  /s  —  0,  .  .  . 


ALGEBKAICALLY    SOLVABLE    EQUATIONS.  287 

This  condition  is  also  sufficient,  as  has  already  been  shown  in 
§§  110,  111,  Theorems  X  and  XII.  Not  that  every  function  belong- 
ing to  G\  on  being  raised  to  the  (px)*^  power  gives  a  function 
belonging  to  G'a  - 1 ;  but  some  function  can  always  be  found  which 
has  this  property,  as  soon  as  the  condition  above  is  fulfilled. 

We  have  then 

Theorem  I.  In  order  that  the  algebraic  equation  f{x)  =  0 
may  be  algebraically  solvable,  it  is  necessary  and  sufficient  that  the 
Jactors  of  composition  of  its  group  should  all  be  prime  numbers. 

§  240.  By  the  aid  of  Theorem  XII,  §  110  we  can  give  this  theo- 
rem another  form 

Theorem  II.  In  order  that  the  algebraic  equation  f{x)  =  0 
may  be  algebraically  solvable,  it  is  necessary  and  sufficient  that  its 
group  should  consist  of  a  series  of  substitutions 

I)  *n  '25  '3j  •  •  •  'f*  ^v  +  i 
which  possess  the  two  following  properties:  1)  the  substitutions  of 
the  group  G\=  jl,  ^1,  ^2?  .  ■  .  t\_^,  tx\  are  commutative,  except  those 
which  belong  to  the  group  G\_i=  \  1,  fi,  ^25  •  •  •  ^x-2)  ^a-iJ,  ccnd  2) 
the  lowest  power  of  t\,  which  occurs  in  G\_i  has  for  its  exponent  a 
prime  number  (cf  also  §  91,  Theorem  XXIV). 

§  241.  Again  the  investigations  of  §  94  enable  us  to  state  The- 
orem I  in  still  a  third  form.  It  was  there  shown  that  if  the  prin- 
cipal series  of  G 

2)  (;,  H,  J,  K,...l 

does  Lot  coincide  with  the  series  of  composition,  then  1)  can  be 
obtained  from  2)  by  inserting  new  groups  in  the  latter,  for  example 
between  H  and  J  the  groups 

H',  li",  .  .  .  Hi^\ 
Then  the  factors  of  composition  which  correspond  to  the  transitions 
from  H  to  H',  from  H'  to  H",  .  .  .  from  H*^)  to  J  are  all  equal. 
Accordingly,  if  all  the  factors  of  composition  belonging  to  1)  are 
are  not  equal,  then  G  has  a  principal  series  of  composition  2). 
We  saw  further  (§95)  that,  if,  in  pagsing  successively  from 
iJ,  H',  H",  ...  to  the  following  group,  the  corresponding  factors  of 
composition  were  all  prime  numbers,  (which  then,  as  we  have  just 


288  THEORY    OF     SUBSTITUTIONS. 

seen,  are  all  equal  to  each  other),  and  only  in  this  case,  the  substitu- 
tions of  H  are  commutative,  except  those  which  belong  to  J.  From 
this  follows 

Theorem  III,  In  order  that  the  algebraic  equation  f(x)  =  0- 
may  be  algebraically  solvable,  it  is  necessary  and  sufficient  that  its 
principal  series  of  composition 

G,  H,  tl,  K,  ...  1 

should  possess  the  property  that  the  substitutions  of  every  group  are 
commutative,  except  those  which  belong  to  the  next  following  group. 

The  substitutions  of  the  last  group  of  the  series,  that  which  pre- 
cedes the  identical  group,  are  therefore  all  commutative. 

§  242.  Before  proceeding  further  with  the  theory,  we  give  a 
few  applications  of  the  results  thus  far  obtained. 

Theorem  IV.  If  a  group  F  is  simply  isomorphic  with  a 
solvable  group  G,  then  F  is  also  a  solvable  group. 

From  §  96  the  factors  of  composition  of  G  coincide  with  those 
of  F.     Consequently  Theorem  IV  follows  at  once  from  Theorem  I. 

Theorem  V.  If  the  group  F  is  multiply  isomorphic  with 
the  solvable  group  G,  and  if  to  the  substitution  1  of  G  corresponds 
the  subgroup  -  of  F,  finally  if  1'  is  a  solvable  group,  then  F  is  also 
solvable. 

The  factors  of  composition  of  F  consist,  from  §96,  of  those  of 
G  and  those  of  -.  Reference  to  Theorem  I  shows  at  once  the 
validity  of  the  present  theorem. 

Theorem  VI.  //  a  group  G  is  solvable,  all  its  subgroups 
are  also  solvable. 

We'write  as  usual 

apply  to  Cj  all  the  substitutions  of  G,  obtain  m?  ^2>  •  •  •  ^r?  and  form 
fir(?)s(l-f,)  {?-?,) . . .  (l-f,.). 

It  is  characteristic  for  the  solvability  of  G  that  g  (?)  can  be  resolved 
into  linear  factors  by  the  extraction  of  roots. 

If  now  H  of  order  r  is  a  subgroup  of  G,  and  if  the  applica- 


ALGEBRAICALLY    SOLVABLE    EQUATIONS.  289 

tion  of  H  to  c,  gives  rise  to  the  values  Cj,  ^3?  •  •  •  ^/j?  then  these 
are  all  contained  among  ^1,  ^2,  .  .  .  ^,..     Consequently 

ft(c)  =  (?- ?,)(?-?,)...  (?-  =  ,) 

is  a  divisor  of  g  (c).  Then  h  (1)  is  also  resolvable  algebraically  into 
linear  factors,  i.  e.,  H  is  a  solvable  group. 

We  might  also  have  proved  this  by  showing  that  all  the  factors 
of  composition  of  H  occur  among  those  of  G. 

Theorem  VIT.  If  the  order  of  a  group  G  is  a  power  of  a 
prime  number  p,  the  group  is  solvable. 

The  group  G  is  of  the  same  type  as  a  subgroup  of  the  group 
which  has  the  same  degree  n  as  G  and  for  its  order  the  highest 
power  p-^  which  is  contained  in  n !  (cf  §  §  39  and  49).  That  the  latter 
group  is  solvable  follows  from  its  construction  (§  89),  all  of  its  fac- 
tors of  composition  being  equal  to  the  prime  number  p.  It  follows 
then  from  Theorem  VI  that  G  is  also  solvable. 

Theorem  VI JI.     //  the  group  G  is  of  order 

r=p,''p/p^yp,^ ... 

where  Pi,P3, Ps, Pi,  .  .  .   are  different  prime  numbers  such  that 

Pi  >  Ih^Ps'^Pi^ .  .  ■  ,     P2>  Pz^Pi^  •  .  • ,     P3  >  i?/  .  .  . , 
then  G  is  solvable.^ 

We  make  use  of  the  theorem  of  §  128,  and  write  r  =  Pi'^q,  where 
then  Pi>  q.  G  contains  at  least  one  subgroup  H  of  the  order  p^\ 
If  we  denote  by  kp^  -\-  1  the  total  number  of  subgroups  of  order  pi" 
contained  in  G^  and  by  PiH  the  order  of  the  maximal  subgroup  of  G 
which  is  commutative  with  H,  then  r  =  p^°-  i{kp^  +  !)•  Since  r  =  p^'^q 
and  q<Pi,  we  must  take  A:  =  0  and  r  =  p^H.  That  is,  G  is  itself 
commutative  with  H.  By  the  solution  of  an  auxiliary  equation  of 
de'gree  q,  with  a  group  of  order  q,  we  arrive  therefore  at  a  function 
belonging  to  the  family  of  H,  and  the  group  G  reduces  to  H(§  232), 
Theorem  X).  From  Theorem  VII  the  latter  group  is  solvable. 
Accordingly,  if  the  auxiliary  equation  is  solvable,  the  group  G  is 
solvable  also. 

The  group  of  the  auxiliary  equation  with  the  order  q=P2^p-^.-. 
admits  of  the  same  treatment  as  G.     Its  solvability  therefore  follows 

*jL.  Sylow:  Math.  Ann.  V,  p.  585. 

19 


290  THEORY    OF    SUBSTITUTIONS. 

from  that  of  a  new  auxiliary  equation  with  a  group  of  order  Ps'^p/ .  .  . , 
and  so  on. 

§  243.     We  return  to  the  general  investigations  of  §  241. 

The  transition  from   G  to  G^  decomposes  the  Galois  resolvent 

T 

equation  into-=Pi  factors.     The  transition  from  G^  to  G^  decom- 

poses  each  of  these  previously  irreducible  factors  inte  —  =  Pa  iiew 
factors,  and  so  on. 

Since  f{x)  =  0  was  originally  irreducible,  but  is  finally  resolved 
into  linear  factors,  it  follows  from  §  235  that  once  or  oftener  a  reso- 
lution of  /(a?)  or  of  its  already  rationally  known  factors  will  occur 
simultaneously  with  the  resolution  of  the  Galois  resolvent  equation 
or  of  its  already  known  rational  factors.  The  number  of  factors 
into  which  f{x)  =  0  resolves,  which  is  of  course  greater  than  1,  must 
from  §  235,  be  a  divisor  of  the  number  of  factors  into  which  the 
Galois  resolvent  equation  divides.  In  the  case  of  solvable  equa- 
tions the  latter  is  always  a  prime  number  jPi,i>2)P3)  •  •  •  Conse- 
quently the  same  is  true  of  f{x)  —  0.  All  prime  factors  of  the 
degree  n  of  the  solvable  equation  f{x)  =  0  are  factors  of  composi- 
tion of  the  group  G,  and  in  fact  each  factor  occurs  in  the  series  of 
composition  as  often  as  it  occurs  in  n. 

To  avoid  a  natural  error,  it  must  be  noted  that  if  in  passing 
from  G  to  G\  the  polynomial  f{x)  resolves  into  rational  factors 
one  of  which  is  f\{x),  this  factor  does  not  necessarily  belong  to  the 
group  G),.  It  may  belong  to  a  family  included  in  that  of  G\. 
The  number  of  values  of  f'\{x)  is  therefore  not  necessarily  equal 
to  rirx.  It  may  be  a  multiple  of  this  quotient.  And  the  product 
f')Sx)  ■  f"\{x)  ...  of  all  the  values  of  f\{x)  is  not  necessarily  equal 
to  f{x),  bat  may  be  a  power  of  this  polynomial. 

We  will  now  assume  that  n  is  not  a  power  of  a  prime  number  p, 
so  that  n  includes  among  its  factors  different  prime  numbers.  Then 
different  prime  numbers  also  occur  among  the  factors  of  composi- 
tion of  the  series  for  G,  and  consequently  (§  94,  Corollary  I)  G  has 

a  principal  series 

G,  H,J,K,..,  M,  1. 

Suppose  that  in  one  of  the  series  of  composition  belonging  to  G 
other  groups 


ALGEBRAICALLY    SOLVABLE    EQUATIONS.  291 

3)  H',  H",  .  .  .  if  W 

occur  between  H  and  J.  Since  n  includes  among  its  factors  at 
least  two  difPerent  prime  numbers,  f{x)  must  resolve  into  factors  at 
least  twice  in  the  passage  from  a  group  of  the  series  of  composition 
to  the  following  one.  Since  the  number  of  the  factors  of  f{x)  is  the 
same  as  the  factor  of  composition,  and  since  the  latter  is  the  same 
for  all  the  intermediate  groups  3),  the  two  reductions  of  f{x)  cannot 
both  take  place  in  the  same  transition  from  a  group  H  of  the  prin- 
cipal series  to  the  next  following  group  J.  It  is  to  be  particularly- 
noticed,  that  all  the  resolutions  of  f{x)  cannot  occur  in  the  transition 
from  the  last  group  M  to  1,  that  is,  within  the  groups 

following  M  in  the  series  of  composition.  At  least  one  of  the  resolu- 
tions must  have  happened  before  M.  Suppose,  for  example,  that  the 
first  resolution  occurs  between  H'  and  H".  Then  it  follows  from 
§  235  that  H'  is  non-primitive  in  those  elements  which  it  connects 
transitively,  and  that  H"  is  intransitive,  the  systems  of  intransitivity 
coinciding  with  the  system  of  non- transitivity  of  H'.  The  same  in- 
transitivity then  occurs  in  all  the  following  groups  H'",  .  .  .  H^^\  and 
likewise  in  the  next  group  J  of  the  principal  series,  which  by  assump- 
tion is  different  from  1.     * 

Suppose  that  J  distributes  the  roots  in  the  intransitive  systems 

1  5  2  5    •    •    •   *^  * )  •^     1  ?  *^     2   •    •    •  *  5  •    •    •        tJU    J     J  *JU    2    5    •    •    •   *^    1    5 

these  systems  being  taken  as  small  as  possible.     Then  the  expression 

f\{oo)  =  (x—x\)  (x  —  x'^)  . .  .  x  —  x\) 
becomes  a  rationally  known  factor  of  /(x),  which  does  not  contain 
any  smaller  rationally  known  factor.     Since  from  the  properties  of 
the  groups  of  the  principal  series 

G-'JG  =  J, 

all  the  values  of  f\{x)  belong  to  the  same  group  J.  They  are  there- 
fore all  rationally  known  with  f\{x).  Of  the  values  of  f\{x)  we 
know  already 

f\{x)  =  (x — x\)     (x — x'2)     .  .  .  {x — x\), 
f\{x)  =  {x—x'\)    {x—x",)     ...(X— a^",), 


fx^^){x)  =  (x  —  x,^"^^)  (ic— ^2^'"))  .  .  .  (a;— aj/*")). 


292  THEORY    OF    SUBSTITUTIONS. 

If  there  were  otker  values,  these  must  have  roots  in  common  with 
some  /a^*^  (x).  Then  /x^*'  (x)  and  consequently  /a(.^')  would  resolve 
into  rational  factors.     This  being  contrary  to  assumption, /\(a?)  has 

71/ 

only  m  =  -   values,  and  is  therefore  a  root  of  an  equation  of  degree 

m.     If  this  equation  is 

4)  ^(2/)  =  (2,-A)(2/_/'\)...(2,-/,C"))  =  0, 
then  f{x)  is  the  result  of  elimination  between  4)  and 

5)  f\{x)  =  af-<pjjj')af-'-^  My')^'~'—  .  . .  =  0, 
where 

My')  =x\-\-x\+x^,-^  ...-^x\. 

V2  \y  )  ^^^  X  -^X  2    \    X  iX  ^  -\-  .  .  .  -\-  X  i  —  iX  ,• , 


so  that  ^1,  4'2^  •  •  •  are  rationally  expressible  in  terms  of  f\.  Since 
f{x)  is  the  eliminant  of  4)  and  5),  it  follows  from  §  228  that  the 
group  of  f{x):=-0  is  non-primitive. 

These  conclusions  rest  wholly  on  the  circumstance  that  J  belongs 
to  the  principal  series  of  G,  and  that  accordingly  G~^J  G  —  J.  It 
is  only  under  this  condition  that  all  the  values  of  f'\{x)  which  occur 
in  the  rational  domain  of  f{x)  =  0  are  rationally  known.  This  shows 
itself  very  strikingly  in  an  example  to  b5  presently  considered. 

Theorem  IX.  //  the  degree  n  of  an  irreducible  algebraic 
equation  is  divisible  by  two  different  prime  numbers,  then  n  can 
always  be  divided  into  two  factors  n  —  im,  such  that  the  given  equa- 
tion f(x)  =  0  resolves  into  m  new  ones 

f',{x) = 0,  r,{x) = 0, . . .  fp{x) = 0, 

which  are  all  of  degree  i,  and  the  coefficients  of  which  are  obtain- 
able from  known  quantities  by  the  solution  of  an  equation  of  degree 
m.*     The  group  of  the  equation  f{x)  =  0  is  non-primitive. 

For  the  purpose  of  comparison  we  consider  the  solution  of  the 
general  equation  of  the  fourth  degree,  to  which,  since  4  =  2^,  the 
preceding  results  are  not  applicable.  It  appears  at  once  that  both 
of  the  resolutions  of  the  polynomial  into  linear  factors  take  place 
in  the  domain  belonging  to  the  last  group  of  the  principal  series 
M,  M\  M",  ...  1.  The  series  of  the  equation  consists  of  the  follow- 
ing groups: 

♦Abel:  Oeuvres  completes  II,  p.  191. 


ALGEBKAICALLY    SOLVABLE    EQUATIONS.  293 

1)  the  symmetric  group; 

2)  the  alternating  group; 

3)  [  1 ,,  {x^  X2)  (xs  X,),  {x,  x^)  (a^2  ^4)  (^1  ^4)  (^2  ^3)  ] ; 

4)  [1,  (a?ia?2)(a?3^4)],  4')  [1 ,  (a^i 0^3)  (a?2 a?^) ],  or 'i")ll,(xiX,)  (x.^x^)]; 

5)  the  group  1. 

The  principal  series  consists  of  the  groups  1),  2),  3),  5).     The  passage 

from  3)  to  4)  and  that  from  4)  to  5)  both  give  the  prime  factor  2. 

The  group  4)  is  the  first  intransitive  one.    For  this /(a?)  resolves  into 

the  two  factors  (x  —  x^)  (x  —  X2)  and  (x — x^)  (x — x^).      But  since 

the  group  4)  does  not  belong  to  the  principal  series,  all  the  six  val 

uesof  (x  —  a?i)  (x — x^)  are  not  known.     If  we  had  chosen  the  group 

4')  instead  of  4),  we  should  have  had  the  two  factors  (x — x^)  (x — x^) 

and  {x — X2){x — x^),  and  so   on.     The  product  of  these  six  values 

give  the  third  power  of  f{x)^{x  —  Xi)  (x — X2)  {x — x^)  {x — x^). 

We  ^an  therefore,  to  be  sure,  resolve  f{x)  into  a  product  of  two 

factors  of  the   second   degree.     But  the   coefficients  of  every  such 

4 
factor  are  the  roots  not  of  an  equation  of  degree  -^  =  2,  but  of   an 

equation  of  degree  6. 

If  we  consider  further  the  irreducible  solvable  equations  of  the 
sixth  degree,  it  appears  that  these  are  of  one  of  two  types,  accord- 
ing as  we  eliminate  y  from 

^'— /i (y)^ 4-/2(2/)  =  0,    y'—c,y-  +  c^y—Cs  =  0, 
or  from 

x'-My)x'  +My)x-My)  =  0,    f  —  c.y-i-e^^O. 

§  244.  The  preceding  results  enable  us  to  limit  our  considera- 
tion to  those  equations  f{x)  =  0  the  degree  of  which  is  a  power  of  a 
prime  number  p.  For  otherwise  the  problem  can  be  simplified  by 
regarding  the  equation  as  the  result  of  an  elimination.  Further- 
more we  may  assume  that  such  a  resolution  into  factors  as  was  con- 
sidered in  the  preceding  Section  does  not  occur  in  the  case  of  our 
present  equations  of  degree  p^,  since  otherwise  the  same  simplifica- 
tion would  be  possible.  We  assume  therefore  that  the  group  of  the 
equation  is  primitive,  thus  excluding  both  the  above  possibilities. 

With  this  assumption  we  proceed  to  the  investigation  of  the 
group.     Suppose  that  the  degree  of  the  equation  is  p^  and  that  its 
principal  series  of  composition  is 
2)  G,  H,J,K,.,.  M,  1, 


294  THEORY    OF     SUBSTITUTIONS. 

In  passing  from  G  through  H,  J,  .  .  .  to  M,  no  resolution  of 
f{x)  into  factors  can  occur.  Otherwise  we  should  have  the  case  of 
the  last  Section,  and  G  would  be  non- primitive.  The  passage  from 
G  to  M  "prepares"  the  equation  f{x)  for  resolution,  but  does  not 
as  yet  resolve  f{x)  into  factors.  The  x  resolutions  of  the  equation 
of  degree  p^  therefore  occurs  in  passing  from  the  last  group  of  the 
principal  series  to  identity,  that  is,  in 

M,  M',  M",  ..  .M''-\  1. 

Accordingly  we  must  have  x^^.  The  application  of  §94,  Corol- 
lary IV  shows  that  all  the  substitutions  of  M  are  commutative. 
The  equation  characterized  by  the  family  of  M  is  therefore  an 
Abelian  equation  of  degree  p^  (§  182).  From  §  94  there  belongs  to 
every  transition  from  one  group  to  the  next  in  the  last  series  the 
factor  of  composition  p,  so  that  the  order  of  M  is  equal  to  p". 
Again  M  can  be  obtained  by  combining  /.  groups  which  have  only 
the  identical  operation  in  common,  which  are  similar  to  each  other, 
and  are  of  order  p.     Suppose  that  these  are 

M(''-'\  M\''-'^  Mi''-'\  .  .  .  Mi^^-^'K 
From  the  above  properties  it  appears  that  every  one  of  these  groups 
is  composed  of  the  powers  of  a  substitution  of  order  p 

and  that  on  account  of  the  commutativity  of  the  groups  (cf.  §  95) 
we  must  also  have 

Sa^  Sp^  =  S/  So!'      («,  /5  =  0, 1,  .  .  .  /  —  1). 
Consequently  every  substitution  of  M  can  be  expressed  by 

S     Sj'^  §2     •   •   •  S  K  —  1  ) 

and  from  the  same  commutative  property 

(s^Si'^S/  .  .  .  S\_,Y  =  S^-PSi'^^Sa"^  .  .  .  sf_i  =  1. 

Every  substitution  of  the  group  M  is  of  order  p.  Our  Abelian 
equation  therefore  belongs  to  the  category  treated  in  §  186,  and  its 
substitutions  are  there  given  in  the  analytic  form 

t^z^,z.^,  .  .  .z^     ^^i  +  aj,  02  +  «2,  .  .  .s^K-f  «k|     (mod._p). 
The  symmetric  occurrence  of   all  the  indices  z^^z-^^  .  .  .z^  already 


ALGEBKAICALLY    SOLVABLE    EQUATIONS.  295 

shows  that  in  the  reduction  of  M  to  1  exactly  x  resolutions  of  the 
polynomial  f{x)  will  occur,  as  is  also  recognized  if  we  write  for 
example 

M'^Zi,Z2,Z.i,  .  .  .z^     Z^,Z.^^a^,Z^-\-a^,...z^-^a^\      (mod.  p), 
M"^Zi,Z2,Zs,  .  .  .z^     2?i,^2,2;3+«3,  .  .  .^k4-«k|  (mod.  p), 

Accordingly  x  =  /,  and  we  have  as  a  first  result 

Theorem  X.  The  last  group  of  the  principal  series  of  a 
primitive,  solvable  equation  of  degree  p"  consists  of  the  p"  arith- 
metic substitutions 

t^Zi,Z2,...z^   2^i  +  «i,2:2+«2j  .  .  •'^/c  +  «k!     (mod.  p), 
the  roots  of  the  equation  being  denoted  by 

^^u^.,...^K    (2:a  =  0,1,2,...p-1). 
Since  (?,  the  group  of  the  equation,  is  commutative  with  M,  it 
follows  from  §  144  that  6r  is  a  combination  of  arithmetic  and  geo- 
metric substitutions.     We  have  therefore  as  a  further  result 

Theorem  XI.  The  group  G  of  every  solvable  primitive  equa- 
tion of  degree  p**  consists  of  the  group  of  the  arithmetic  substitu- 
tions of  the  degree  p",  combined  with  geometric  substitutions  of  the 
same  degree 

u^Zi,  z.2,...z^   a^Zi-\-b,Z2  +  .-.-\-CiZ^,aoZ,-{-h2Z2 -{-... -i-c.,z^,...\ 

(mod.  p). 

§  245.  Before  proceeding  further  with  the  general  investigation, 
we  consider  particularly  the  cases  x  =  1  and  x  =  2,  the  former  of 
which  we  have  already  treated  above. 

We  consider  first  the  solvable,  primitive  equations  of  prime  de- 
gree p.  We  may  omit  the  term  "primitive,'*  since  non  primitivity 
is  impossible  with  a  prime  number  of  elements. 

The  group  of  the  most  general  solvable  equation  of  degree  p  must 
then  coincide  with  or  be  contained  in 

G^\z  az-\-a\  (a  =  l,  2,  ...p  —  1;  a  — 0,1,  ...  p  —  1)     (mod.p). 

We  prove  that  the  former  is  the  case,  by  constructing  the  groups  of 
composition  from  G  to  M  and  showing  that  all  the  factors  of  com- 


296 


THEORY    OF    SUBSTITUTIONS. 


position  which  occur  are  prime  numbers.     We  divide  p  —  1  into  its 
prime  factors:  p — 1  ==  ^1^2  •  •  •  ?  and  construct  the  subgroup 

p  —  1 


if  ^  I  z  a,q,  0  +  «i  I  {a,  =  1,  2,  . 
then  the  subgroup 

J^\z  a2^i^2^  +  «2|  (a2  =  l,  2, .. 


0,1,... p-1), 


;  «2  =  0,  1,  ..  .p  —  1), 


and  so  on.     Then  H,  J,  .  .  .    all  belong  to  the  principal  series  of  G. 
Thus  we  have,  for  example  G~^JG  —  J.    For,  if  we  take 


then 


t^ 

2;   az-\-a\, 

^-^^ 

. !(.-«) 

? 

-a)     . 

z   a^q.q^z-^-a^    . 

2;   a2;  +  « 

0    a,q,q,z- 

,   a2  9'ig'2«  — «+"2 

"^ 

■^ 

, 

so  that  the  transformation  of  a  substitution  of  J  with  respect  to  any 
substitution  of  G  leads  to  another  substitution  of  J.  Evidently  the 
principal  series  coincides  here  with  the  series  of  composition.  The 
factors  of  composition  ^i ,  ^2 »  •  •  •  are  all  prime  numbers.  The  proof 
is  then  complete. 

If  a  substitution  of  ,G  leaves  two  roots  Xx  and  x^  unchanged,  then 
it  leaves  all  the  roots  unchanged.  For  from  /  =  ak  -(-  a ,  />-  =  ait.  +  « 
follows  necessarily  a::^!,  «i^0,  and  the  substitution  becomes  iden- 
tical: \  —  \z   z\. 

If  a  substitution  of  G  leaves  one  root  Xx  unchanged  and  if  it  con 
verts  a?A  +  i  into  x^,  then  every  x^,  becomes  a?(^_A.)(v-A)  +  A-     For  from 
A  ^a  /  +  « , !'-  ^  a(A  +  1)  +  a ,  follows  a^//  —  / ,  a  ^ /(/ — jj.  —  1),  and 
the  substitution  is  of  the  form  \z    ( //  —  A)z-]r'^^  Q- ■-:>■  +  0  !  • 

If  a  substitution  of  G  leaves  no  root  unchanged,  and  if  it  con- 
verts Xx  into  a?^,  then  every  Xy  is  converted  into  x^j^^^^x-  For  only 
in  this  case  is  there  no  solution  I  of  the  congruence  /^^ax-l-a, 
when  aEE;  1.  If  /  +  1  is  to  become  ,a,  then  we  must  have  /j-.  =  A  +  a. 
This  gives  a  =  // — A,  and  the  substitution  is  |  2:  z-\-  ji.  —  >^^  | . 

These  are  precisely  the  same  results  which  the  earlier  algebraic 
method  furnished  us. 


ALGEBBAICALLY    SOLVABLE    EQUATIONS.  297 

Theorem  XII.  The  general  solvable  equations  of  prime 
degree  p  are  those  of  §  196.  Their  group  is  of  order  p(p  —  1)  and 
consists  of  the  substitutions  of  the  form 

s^\z  az-\-a\    (a  =  l,  2,  ...p — 1;  a  =  0, 1,  .  .  .p — 1)     (mod.  p). 

Its  factors  of  composition  are  all  prime  divisors  of  p  —  1,  each  fac- 
tor ocurring  as  many  times  as  it  occurs  in  p  —  1,  and  beside  these 
p  itself. 

§  246.  We  pass  to  the  general  solvable  primitive  equations  of 
degree  p^     As  a  starting  point  we  have  the  arithmetic  substitutions 

t=\Zi,Z2   2:,  +  «i,2?2  +  «2|     (mod.p), 

which  form  the  last  group  M  of  the  corresponding  principal  series. 
To  arrive  at  the  next  preceding  group,  we  must  determine  a  substi- 
tution s  which  has  the  following  properties.     Its  form  is 

s^\zi,Z2   ai2ri-f-^i'2^2)  «2^i  +  ^2  2^2 1     (mod.p), 

and  the  lowest  power  of  s  which  occurs  in  M,  and  is  therefore  of 
the  form  t,  must  have  a  prime  number  as  exponent.  Since  now  all 
the  powers  of  s  are  of  the  same  form  as  s  itself,  the  required  power 
must  be  \zi,  z.2  ^i ,  2^2 1  =  1.  That  is,  the  order  of  the  substitution 
s  must  be  a  prime  number. 

From  these  and  other  similar  considerations  we  arrive  at  the  fol- 
lowing results,  *  the  further  demonstration  of  which  we  do  not  enter 
upon. 

Theorem  XIII.  The  general  solvable,  primitive  equations 
of  degree  p^  are  of  three  different  types. 

The  first  type  is  characterized  by  a  group  of  order  2p^(p  —  1)^, 
the  substitutions  of  which  are  generated  by  the  following : 

\z,,Z,    Z,  +  a^,   Z,-{-a,\       (ai,a2  =0,1,  2,..  .p  —  1), 

I  /  100  i\      (mod.  p), 

\Zi,Z2   a.z.^a^z^]  (ai,  a2,  =  l,  2,  3, .  .  .p  — 1), 

pl  »  ^2     ^2  ?  2^1  I . 

The  groups  belonging  to  the  second  type  are  of  order  2  p^i^p^ — 1), 
and  their  substitutions  are  generated  by  the  following : 

•C.  Jordan:  Liouville,  Jour,  de  Miitb.  (2)  XIII,  pp.  111-J35, 

20 


298  THEOBY    OF    SUBSTITUTIONS. 

\Zi,Z^    Zi-\-a,,z,-\-a,\  («,,  a2=0,l,  2,  ...p— 1), 

\zi^Z2   aZi-{- bez2<, bz^  +  az<i\     (a. 6^0, 1,  .  .  . p — 1;  hut  not  a,  5^0), 

\zuZ2   z^,—Z2\,  (mod.p). 

where  e  is  any  quadratic  remainder  (mod.  p). 

The  groups  of  the  third  type  are  of  order  24p^(p  —  1).  The 
form  of  their  substitutions  is  different,  according  as  p^l  or  p^3 
(mod.  4).  In  the  former  case  the  group  contains  beside  the  two 
substitutions 

\Zx,Z2    Z,-\-a,^   z^  +  a^l      (ai,a2  =  0,  1,2,  ...p  — 1),      /     ^^      x 

\Zi,Z2   azi,az2\  (a^l,  2,  3  .  .  .p  — 1), 

also  the  following  four: 

I  01 ,  2^2     IZi ,  IZ2  I  ,  I  01  ,  2^2     *^2  J  *^1  I  » 

I  01  ,  02     Zi  ^02 ,  01  -}-  102  I  ,        I  0]  )  02     Zi-\~  Z2y  Zi  02  I  , 

where  i  is  a  root  of  the  congruence  i^^  —  1  {mod.  p).  If  p^S 
(mod.  4),  the  group  contains  the  first  two  substitutions  above, 
together  with  the  following  four: 

I  0 1 ,  02  02 ,     01  I  ,  \  Z\,  Z2     SZi   -j~  C02 ,  C01    S02  I  , 

|0i,02   —{l  +  st)z,-}-(s—f)z2,  {t-\-s'')z,-{-{st—s—t)z2\, 
|0i,02  S0i  +  (1+O^2,  (*— l)2;i— S02I, 

where  s  and  t  satisfy  the  congruence  s'^-\-f^ — 1  {mod.  p). 

For  p  =  3  the  first  and  second  types,  and  for  p  =  b  the  second 
type  are  not  general.  These  types  are  then  included  as  special 
cases  in  the  third  type,  which  is  always  general. 

§  247.  We  return  from  the  preceding  special  cases  to  the  more 
general  theory. 

The  same  method  which  we  have  employed  above  in  the  case  of 
p^  can  be  applied  in  general  to  determine  the  substitutions  of  the 
group  L  which  precedes  M  in  the  principal  series  of  composition. 
L  is  obtained  by  adding  to  the  substitutions 

«=|0i,02,  ...0K     0,  +  «i,02-f«2,  ..  .'S'k  +  ^k)     (inod.p) 
of  Jf  a  further  substitution 

^^|0p0g,  .  .  .0K  ai0,+  6,02+..  .+Ci0«,a20j+V2+---+C2^K,  ...  I 

(wod,  p)f 


ALOEBEAICALLY  SOLVABLE  EQUATIONS.  299 

where  the  first  power  of  s  to  occur  among  the  fa  has  a  prime  expo 
nent.     Since  all  the  powers  of  s  are  of  the  same  form  as  s  itself, 
any  power  of   s  which  occurs  among  the  f  s  must  be  equal  to  1. 
Consequently  s  must  be  of  prime  order.      It  is  further  necessary 
the  group  L  =  \t,  s\  should  not  become  non-primitive. 

§  248.     From  the  form  to  which  the  substitutions  of  G  are  re- 
stricted, we  have  at  once 

Theorem  XIV.     All  the  substitutions,  except  identity,  which 
belong  to  the  group 

affect  all  the  elements. 

The  converse  proposition,  which  was  true  for  z  =  1,  does  not  hold 
in  the  general  case.     For  the  element  a-^j ,  ^r, .  •  •  «*c  i^  unaffected  by 

s=\Zi,Z2,...z  ai2;i+Mi+-..+  Ci2;«+«u  a22?i+622;2+-..+  C22^2  +  «2,- I 
only  in  case  the  x  congruences 

(ai— 1)2:1+  612:2+...+  Ci2f^  +  ai^0, 

^^  a,,,  +  (5,_l).,+  ...+  C2..  +  a2=0,    ^^^^^^ 

a^Zi-i-  b^Z2+  ...-\-{c^—l)z^-{-a^=0 

are  satisfied.     Consequently,  as  soon  as  the  determinant 
tti  —  1  fei         .  .  .  Ci 


1    .    .    .Co 


^0     (mod.  p), 


a^         b^         ...c^  —  1 

the  oci,  ttg,  . .  .  a^  can  be  so  chosen  that  the  congruences  S)  are  not 
satisfied  by  any  system  Zi,Zci,  .  .  .z^. 

We  consider  now  all  the  substitutions  of  the  group  G  which 
leave  one  element  unchanged.  Since  the  distinction  between  the 
elements  is  merely  a  matter  of  notation,  we  may  regard  a?o,o,...o  as 
the  fixed  element.  Then  the  substitutions  which  leave  this  element 
unchanged  are 

r^z^,z^,...z^  a^z^-\-b,Z2-\-...-\-c^z^,  a^z^-\-br,Z2  +  ...  +  CiZ^...\. 
If  we  adjoin  iCo.c.o  to  the  equation,  the  group  G  reduces  to  F. 


300  THEORY    OF    SUBSTITUTIONS. 

Since  all  tlie  substitutions  of  G  are  obtained  by  appending  to  those 
of  r  the  constants  a^,  «2)  •  •  •  and  since  the  a's  can  be  chosen  in  p* 
ways,  it  follows  that  the  adjunction  of  a  single  root  reduces  G  to  its 
{p^Y"  part. 

§  249.  We  will  now  consider  the  possibility  that  a  substitution 
of  G  leaves  x  + 1  elements  ^^i , ^2 , •  •  .^^  unchanged.  Then  the  con- 
gruences S)  of  the  preceding  Section  are  satisfied  by  x  +  1  systems 
of  values  z-^^z^, .  . .  Zk 

z,  =  C,^^\  z,  =  C2^^),  ...z„  =  :.<^\      (A  =.  0, 1,  2, . . .  x). 

We  will  howeyer  regard  not  the  coefficients  a,  6,  ...  c;  a  of  the 
substitution  but  the  values  C/^\  ^2^^^  •  •  •  ^S^^  as  known,  and  attempt 
to  determine  the  substitution  from  these  data.  If  now  the  determ- 
inant 


E  = 


r'      r'  r' 

>»     1  -»     2         .     .     •     S     A 


is   not  ^0  (mod.  p),  then  the  x  systems    Tj),  Tg),  .  .  .  T^)    each  of 
X -|-  1  congruences  with  the  unknown  quantities  a,b,  , .  .  c;  a 

T,)  {a,  —  1)  C/^)  +  &i  C^^"^  + . . .  +  c,  C.(^)  +  «i =0, 

T.)  a. C/^)  +  &. C/")  + . . .  +  (c.- 1) C.W  +  «<^0, 

have  only  one  solution  each,  viz: 

Li)  «!  =  1,  &i  =  0,  .  . .  Cj  =  0;  «i  =  0, 

La) '  ttg  =  0,  62  =  Ij  .  • .  C2  =  0;  flg  =  Oj 


jL^)  CTk^O,  6^  =  0, .  ..c«  =  1;  aK  =  0, 

and  these  solutions  furnish  together  the  identical  substitution  1. 

We  designate  now  a  system  of  x-\-l  roots  of  an  equation  for 
which  E^O  (mod.  p)  as  a  system  of  conjugate  roots.  • 

We  have  then 

Theorem  XV.  If  a  substitution  of  a  primitive  solvable 
group  of  degree  p"  leaves  unchanged  x  -f- 1  roots  which  do  not 
form  a  conjugate  system,  the  substitution  reduces  to  identity. 


■»   1     *  2        • 

ft 

ftt     fit 

■»     1   ^     2      • 

ftt 

:(")  c/*^) ; 

. .  c('>. 

ALOEBHAICALLY    SOLVABLE    EQUATIONS.  301 

If  therefore  we  adjoin  ^  + 1  such  roots  to  the  equation,  the 
group  G  reduces  to  those  substitutions  which  leave  y.-\-\  roots 
unchanged,  i.  e.,  to  the  identical  substitution.  The  equation  is  then 
solved. 

Theorem  XVI.  All  the  roots  of  a  solvable  primitive  equa- 
tion of  degree  p"  can  be  rationally  expressed  in  terms  of  any  y.-\-l 
among  them,  provided  these  do  not  form  a  conjugate  system. 

If  we  choose  the  notation  so  that  one  of  the  ^  +  1  roots  is 
^0 ,0, ...  0,  the  determinant  becomes 


±E. 


If  the  roots  are  not  to  form  a  conjugate  system,  then  ^  =  0 
(mod.  p).  The  number  r  of  systems  of  roots  which  satisfy  this  con- 
dition is  determined  in  §  146.     We  found 

r  =:  (p"  — 1)  (p^—p)  (p^—p')  .  .  .  (p^—p"'^). 
Theorem  XVII.     For  every  root  ^«i,^2,...^^  ^^  (^(^'^  deter- 
mine 

(p'^  —  l)  (p'^—p) .  .  .  jp'—p"-') 
1,  2,  ...  X 

systems  of  x  roots  each  such  that  these  x-\-l  roots  do  not  form  a  con- 
jugate system,  so  that  all  the  other  roots  can  be  rationally  expressed 
in  terms  of  them.     The  system  composed  of  the  z  -f- 1  roots 

•^'O  ,  0  ,  0  ,  . .  .  >       "^l ,  0  , 0  . .  .  0  >       "^O  ,  1  ,  0  , .  .  0  )    ...    '^0,0,0,...! 

is  appropriate  for  the  expression  of  all  the  roots. 

These  results  throw  a  new  light  on  our  earlier  investigations  in 
regard  to  triad  equations,  in  particular  on  the  solution  of  the  Hes- 
sian equation  of  the  ninth  degree  (cf  §  §  203-6).  It  is  plain  that 
we  can  construct  in  the  same  way  quadruple  equations  of  degree 
p^,  and  so  on. 


^DOK    MU5T  SB 
CHA-RGEP  OOT  WITH 
CASE 


>V' 


14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

LOAN  DEPT. 

This  book  is  due  on  the  last  date  stamped  below,  ot 

on  the  date  to  which  renewed. 

Renewed  books  are  subject  to  immediate  recall. 


Ri£<.0  LD 


AUG  U '65 -12  M 


PE.B   5^96854 


N3\r2ni973  4  7 


m  fiQ\2^''^'^^*' 


LD  21A-60m-4,'64 
(E4555sl0)476B 


General  Library 

University  of  California 

Berkeley 


P.?of?ir?5 


QA  17 


THE  UNIVERSITY  OF  CAUFORNIA  UBRARY 


.    «y_*.        t 


